Non-local operators, Pseudodifferential Equations of Parabolic Type With Variable Coefficients and Markov Processes over p-adics

◆♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs✱ Ps❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ♦❢

P❛r❛❜♦❧✐❝ ❚②♣❡ ❲✐t❤ ❱❛r✐❛❜❧❡ ❈♦❡✣❝✐❡♥ts ❛♥❞ ▼❛r❦♦✈

Pr♦❝❡ss❡s ♦✈❡r

p

−

❛❞✐❝s

▲❡♦♥❛r❞♦ ❋❛❜✐♦ ❈❤❛❝ó♥ ❈♦rtés

▼❛t❡♠át✐❝♦

❈❡♥tr♦ ❞❡ ■♥✈❡st✐❣❛❝✐ó♥ ② ❞❡ ❊st✉❞✐♦s ❆✈❛♥③❛❞♦s ❞❡❧

■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ◆❛❝✐♦♥❛❧

❈✐♥✈❡st❛✈✲■P◆

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛s

▼é①✐❝♦✱ ❉✳❋✳

◆♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs✱ Ps❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ♦❢

P❛r❛❜♦❧✐❝ ❚②♣❡ ❲✐t❤ ❱❛r✐❛❜❧❡ ❈♦❡✣❝✐❡♥ts ❛♥❞ ▼❛r❦♦✈

Pr♦❝❡ss❡s ♦✈❡r

p−

❛❞✐❝s

▲❡♦♥❛r❞♦ ❋❛❜✐♦ ❈❤❛❝ó♥ ❈♦rtés

▼❛t❡♠át✐❝♦

❚❤❡s✐s ❲♦r❦ t♦ ❖❜t❛✐♥ t❤❡ ❉❡❣r❡❡ ♦❢

❉♦❝t♦r ❡♥ ❈✐❡♥❝✐❛s✱ ▼❛t❡♠át✐❝❛s

❆❞✈✐s♦r

❉r✳ ❲✐❧s♦♥ ❩úñ✐❣❛ ●❛❧✐♥❞♦

■♥✈❡st✐❣❛❞♦r ❚✐t✉❧❛r ✸❈ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛s

❈❡♥tr♦ ❞❡ ■♥✈❡st✐❣❛❝✐ó♥ ② ❞❡ ❊st✉❞✐♦s ❆✈❛♥③❛❞♦s ❞❡❧ ■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ◆❛❝✐♦♥❛❧

▼é①✐❝♦

❈❡♥tr♦ ❞❡ ■♥✈❡st✐❣❛❝✐ó♥ ② ❞❡ ❊st✉❞✐♦s ❆✈❛♥③❛❞♦s ❞❡❧

■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ◆❛❝✐♦♥❛❧

❈✐♥✈❡st❛✈✲■P◆

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛s

▼é①✐❝♦✱ ❉✳❋✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t

❋♦r❡♠♦st✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② s✐♥❝❡r❡ ❣r❛t✐t✉❞❡ t♦ ♠② ❛❞✈✐s♦r ❉r✳ ❲✐❧s♦♥ ➪❧✈❛r♦ ❩úñ✐❣❛ ●❛❧✐♥❞♦ ❢♦r ❤✐s ❝♦♥t✐♥✉♦✉s s✉♣♣♦rt ❢♦r ♠② ❞♦❝t♦r❛❧ st✉❞✐❡s ❛♥❞ r❡s❡❛r❝❤✱ ❢♦r ❤✐s ♣❛t✐❡♥❝❡✱ ♠♦t✐✈❛t✐♦♥✱ ❡♥t❤✉s✐❛s♠✱ ❛♥❞ ✐♠♠❡♥s❡ ❦♥♦✇❧❡❞❣❡✳ ❍✐s ❣✉✐❞❛♥❝❡ ❤❡❧♣❡❞ ♠❡ ✐♥ ❛❧❧ t❤❡ t✐♠❡ ❞✉r✐♥❣ ♠② ❞♦❝t♦r❛❧ r❡s❡❛r❝❤ ❛♥❞ ✇r✐t✐♥❣ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥✳ ■ ❝♦✉❧❞ ♥♦t ❤❛✈❡ ✐♠❛❣✐♥❡❞ ❤❛✈✐♥❣ ❛ ❜❡tt❡r ❛❞✈✐s♦r ❛♥❞ ♠❡♥t♦r ❢♦r ♠② P❤✳❉ st✉❞✐❡s✳ ■ ❛❧s♦ ✇✐s❤ t♦ t❤❛♥❦ t♦ ❉r✳ ❙❡r❣✐✐ ❚♦r❜❛ ❢♦r ♠❛♥② ❢r✉✐t❢✉❧ ❞✐s❝✉ss✐♦♥s ❛♥❞ r❡♠❛r❦s ❞✉r✐♥❣ t❤❡ s❡♠✐♥❛r ♦♥ p−❛❞✐❝ ♥✉♠❜❡rs✱ ✉❧tr❛♠❡tr✐❝ ❛♥❛❧②s✐s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❛❦❡ t❤✐s t✐♠❡

❈♦♥t❡♥ts

❈♦♥t❡♥ts ■

❚❤❡s✐s ❖✈❡r✈✐❡✇ ■■■

✶✳ p−❛❞✐❝ ❆♥❛❧②s✐s ✶

✶✳✶ ❚❤❡ ✜❡❧❞ ♦❢ p✲❛❞✐❝ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❚❤❡ ❇r✉❤❛t✲❙❝❤✇❛rt③ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✹ ❊❧❧✐♣t✐❝ Ps❡✉❞♦ ❉✐✛❡r❡♥t✐❛❧ ❖♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷✳ P❛r❛❜♦❧✐❝✲t②♣❡ ❊q✉❛t✐♦♥s ❛♥❞ ❯❧tr❛♠❡tr✐❝ ❘❛♥❞♦♠ ❲❛❧❦s ✹

✷✳✶ ❆ ◆❡✇ ❈❧❛ss ♦❢ ◆♦♥❧♦❝❛❧ ❖♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❙♦♠❡ ❛❞❞✐t✐♦♥❛❧ r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ p✲❛❞✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ r❡❧❛t✐♦♥ ✐♥ ❝♦♠♣❧❡① s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ❍❡❛t ❑❡r♥❡❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✺ ▼❛r❦♦✈ Pr♦❝❡ss❡s ♦✈❡r Qn_p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✻ ❚❤❡ ❈❛✉❝❤② Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✻✳✶ ❍♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ✐♥S ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✻✳✷ ❍♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ✐♥L2_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹}

✷✳✻✳✸ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✼ ❋✐rst P❛ss❛❣❡ ❚✐♠❡ Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✸✳ ❋✐rst P❛ss❛❣❡ ❚✐♠❡ ✷✶

✸✳✶ Pr❡❧✐♠✐♥❛r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ▼❛r❦♦✈ Pr♦❝❡ss❡s ♦✈❡r Q4_p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸✳✸ ❚❤❡ ❋✐rst P❛ss❛❣❡ ❚✐♠❡ ♦✈❡r Q4_p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

❈❖◆❚❊◆❚❙ ■■

✸✳✹ ❚❤❡ ❋✐rst P❛ss❛❣❡ ❚✐♠❡ ♦✈❡r Q2_p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✺ ❙✉r✈✐✈❛❧ ♣r♦❜❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✹✳ ◆♦♥✲❆r❝❤✐♠❡❞❡❛♥ P❛r❛❜♦❧✐❝✲t②♣❡ ❊q✉❛t✐♦♥s ✸✸

✹✳✶ ❆ ❝❧❛ss ♦❢ ♥♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ P❛r❛❜♦❧✐❝✲t②♣❡ ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✷✳✶ ❈❧❛✐♠u(x, t)∈M_λ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹✳✷✳✷ ❈❧❛✐♠u(x, t)s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✷✳✸ ❈❧❛✐♠u(x, t)✐s ❛ s♦❧✉t✐♦♥ ♦❢ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✭✹✳✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✸ P❛r❛❜♦❧✐❝✲t②♣❡ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✸✳✶ P❛r❛♠❡tr✐③❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸✳✷ ❍❡❛t ♣♦t❡♥t✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✸✳✸ ❈♦♥str✉❝t✐♦♥ ♦❢ ❛ s♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✹ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✺ ▼❛r❦♦✈ Pr♦❝❡ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✻ ❚❤❡ ❈❛✉❝❤② Pr♦❜❧❡♠ ✐s ❲❡❧❧✲P♦s❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❚❤❡s✐s ❖✈❡r✈✐❡✇

■♥tr♦❞✉❝t✐♦♥

❉✉r✐♥❣ t❤❡ ❧❛st t✇❡♥t②✲✜✈❡ ②❡❛rs t❤❡r❡ ❤❛s ❜❡❡♥ ❛ str♦♥❣ ✐♥t❡r❡st ♦♥p−❛❞✐❝ ❛♥❛❧②s✐s ❞✉❡

t♦ ✐ts ❝♦♥♥❡❝t✐♦♥s ✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✱ s❡❡ ❡✳❣✳ ❬✸✷❪✱ ❬✸✵❪✱ ❬✹❪✱ ❬✻❪✱ ❬✺❪✱ ❬✾❪✱❬✼❪✱ ❬✷✵❪ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♥❡✇ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s ❤❛✈❡ ❛r✐s❡♥✱ ❛♠♦♥❣ t❤❡♠✱ t❤❡ st✉❞② ♦❢ ♥♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♣❛r❛❜♦❧✐❝✲t②♣❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡✳

❚❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s ♦r❣❛♥✐③❡❞ ✐♥t♦ t❤r❡❡ ♣❛rts✳ ■♥ t❤❡ ✜rst ♣❛rt✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ t②♣❡ ♦❢ ♥♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs ❛♥❞ st✉❞② t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r ❝❡rt❛✐♥ ♣❛r❛❜♦❧✐❝✲t②♣❡ ♣s❡✉✲ ❞♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ♦♣❡r❛t♦rs✳ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦✲ ❧✉t✐♦♥s ♦❢ t❤❡s❡ ♣❛r❛❜♦❧✐❝✲t②♣❡ ❡q✉❛t✐♦♥s ❛r❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ♦♥ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r t❤❡ ✜❡❧❞ ♦❢p✲❛❞✐❝ ♥✉♠❜❡rs✳ ❲❡ st✉❞② s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ r❛♥❞♦♠ ✇❛❧❦s✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❡ st✉❞② t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ❝❡rt❛✐♥ ❡❧❧✐♣t✐❝ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ✐♥ ❞✐♠❡♥s✐♦♥s4❛♥❞2♦✈❡r t❤❡p✲❛❞✐❝s✳ ❋✐♥❛❧❧②✱ ✐♥ t❤❡ t❤✐r❞ ♣❛rt✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ❝❧❛ss

♦❢ ♣❛r❛❜♦❧✐❝✲t②♣❡ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts ♦✈❡r t❤❡ p✲❛❞✐❝s✳ ❲❡ ❡st❛❜❧✐s❤ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s ❢♦r t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥♥❡❝t❡❞ ✇✐t❤ ▼❛r❦♦✈ ♣r♦❝❡ss❡s✳ ❚❤❡s❡ r❡s✉❧ts ✇❡r❡ ♦❜t❛✐♥❡❞ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♠② ❛❞✈✐s♦r ❉r✳ ❲✳❆✳ ❩úñ✐❣❛✲●❛❧✐♥❞♦ s❡❡ ❬✶✽❪✱❬✶✾❪ ❛♥❞ ❬✶✼❪✳

❙t❛t❡ ♦❢ t❤❡ ❆rt

▲❡t p ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡ ✜❡❧❞ ♦❢ p−❛❞✐❝ ♥✉♠❜❡rs Q_p ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❝♦♠♣❧❡t✐♦♥

♦❢ t❤❡ ✜❡❧❞ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs Q ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ p−❛❞✐❝ ♥♦r♠ |·|_p✳ ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t Qn_p ✐s ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t t♦♣♦❧♦❣✐❝❛❧ ✜❡❧❞✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠

F ✐s ❛✈❛✐❧❛❜❧❡ ♦♥ Qn_p✳ ❇② ✉s✐♥❣ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ♣♦ss✐❜❧❡ t♦ ✐♥tr♦❞✉❝❡ p−❛❞✐❝

♣s❡✉❞♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✳ ❋♦r♠❛❧❧② s♣❡❛❦✐♥❣✱ ❛ ♣s❡✉❞♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❤❛s t❤❡ ❢♦r♠F−1((symbol)Fϕ)✱ ✇❤❡r❡ ❢✉♥❝t✐♦♥ ϕ✐s ✐♥ ❛ s✉✐t❛❜❧❡ s♣❛❝❡✳

❚❍❊❙■❙ ❖❱❊❘❱■❊❲ ■❱

❚❤❡ p−❛❞✐❝ ❍❡❛t ❊q✉❛t✐♦♥s✳

❚❤❡ ♠♦st ❜❛s✐❝p−❛❞✐❝ ❤❡❛t ❡q✉❛t✐♦♥s ❤❛✈❡ t❤❡ ❢♦r♠

∂u(x, t)

∂t + (D

α_{u)(x, t) = 0,}

✇❤❡r❡ t > 0✱ x ∈ Q_p ❛♥❞ (Dαϕ)(x, t) = F_ξ−_→1_x(|ξ|α_p Fx→ξϕ(x, t))✱ α > 0 ✐s t❤❡ ❱❧❛❞✐♠♦✈

❖♣❡r❛t♦r✳ ❚❤❡s❡ ❡q✉❛t✐♦♥s ❛r❡ t❤❡ p−❛❞✐❝ ❝♦✉♥t❡r♣❛rts ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ❤❡❛t ❡q✉❛t✐♦♥s✱

❢♦r ✐♥st❛♥❝❡✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ ♦❢ ❡q✉❛t✐♦♥s ❛r❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ♦✈❡r Q_p✱ s❡❡ ❡✳❣✳ ❬✸✸❪✳

❚❤❡p−❛❞✐❝ ♣s❡✉❞♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ t❤❡✐r ❝♦♥♥❡❝t✐♦♥s ✇✐t❤ ▼❛r❦♦✈ ♣r♦❝❡ss❡s

❤❛✈❡ ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞✳ ■♥ ❬✷✺❪✱ ❬✷✹❪ ❑✉❝❤✉❜❡✐ st✉❞✐❡❞ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❱❧❛✈✐♠✐r♦✈ ♦♣❡r❛t♦r✳ ❍❡ s❤♦✇❡❞ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ st✉❞✐❡❞ t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss❡s✳ ■♥ ❬✷✽❪ ❏✳❏✳ ❘♦❞rí❣✉❡③✲❱❡❣❛ ❛♥❞ ❲✳❆✳ ❩úñ✐❣❛✲●❛❧✐♥❞♦ ❣❡♥❡r❛❧✐③❡❞ s♦♠❡ ♦❢ ❑♦❝❤✉❜❡✐✬s r❡s✉❧ts t♦ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s②♠❜♦❧k·kα_p ✭t❤❡ ❚❛✐❜❧❡s♦♥ ♦♣❡r❛t♦r✮✳ ■♥ ❬✸✹❪ ❲✳❆✳ ❩úñ✐❣❛✲●❛❧✐♥❞♦ ❝♦♥s✐❞❡r❡❞ t❤❡

❈❛✉❝❤② ♣r♦❜❧❡♠ ❛ss♦❝✐❛t❡❞ t♦ ❛ ♣s❡✉❞♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✇✐t❤ s②♠❜♦❧ |f(ξ)|β_p✱ ✇❤❡r❡

β > 0 ❛♥❞ f(x) ∈ Q_p[ξ1,· · · , ξn] ✐s ❛ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡d t❤❛t ✈❛♥✐s❤❡s

♦♥❧② ❛t t❤❡ ♦r✐❣✐♥✳ ❍❡ s❤♦✇❡❞ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠✱ ❛♥❞ ❛❧s♦ st✉❞✐❡❞ t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss ❛tt❛❝❤❡❞ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥s✳

P❤②s✐❝❛❧ ♠♦t✐✈❛t✐♦♥s

❚❤✐s t❤❡s✐s ✐s ♠♦t✐✈❛t❡❞✱ ♦♥ t❤❡ ♣❤②s✐❝❛❧ s✐❞❡✱ ❜② ❍❛♥s ❋r❛✉❡♥❢❡❧❞❡r✬s ❝♦♥❥❡❝t✉r❡ ❛❜♦✉t t❤❡ ✉❧tr❛♠❡tr✐❝ ♥❛t✉r❡ ♦❢ ♣r♦t❡✐♥s✳ ❆✈❡t✐s♦✈ ❡t ❛❧✳ ❤❛✈❡ s❤♦✇❡❞ t❤❛t ❋r❛✉❡♥❢❡❧❞❡r✬s ❝♦♥❥❡❝t✉r❡ ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ♠❛t❤❡♠❛t✐❝❛❧❧② ✉s✐♥❣p−❛❞✐❝ ❛♥❛❧②s✐s✱ ✐♥ ❛❞❞✐t✐♦♥✱ t❤❡② ♣r♦♣♦s❡❞ ♥❡✇

p−❛❞✐❝ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❢♦r ♣r♦t❡✐♥ ❞②♥❛♠✐❝s✱ s❡❡ ❡✳❣✳ ❬✻❪✱ ❬✺❪✱ ❬✽❪✱ ❬✶✵❪✱ ❬✶✷❪✳ ❖♥ t❤❡

♠❛t❤❡♠❛t✐❝❛❧ s✐❞❡✱ ✇❡ ✐♥t❡♥❞ t♦ ❝♦♥t✐♥✉❡ ❛♥❞ ❡①t❡♥❞ t❤❡ ✇♦r❦ ♦❢ ❆✳◆✳ ❑♦❝❤✉❜❡✐ ❛♥❞ ❲✳❆✳ ❩úñ✐❣❛✲●❛❧✐♥❞♦ ♦♥ p−❛❞✐❝ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ♣❛r❛❜♦❧✐❝✲t②♣❡✱ s❡❡ ❡✳❣✳ ❬✷✺❪✱

❬✷✹❪✱ ❬✷✽❪✱ ❬✸✹❪ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳

❍❛♥s ❋r❛✉❡♥❢❡❧❞❡r ❡t ❛❧✳ ❡st❛❜❧✐s❤❡❞ ❡①♣❡r✐♠❡♥t❛❧❧② ✐♥ ❬✷❪✲❬✸❪ t❤❛t ♣r♦t❡✐♥s ❡①❤✐❜✐t ❛ ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡✳ ■♥ ❋✐❣✉r❡ ✶✳❜✱ t❤❡ t♦♣ ✈❡rt❡① r❡♣r❡s❡♥ts t❤❡ s♣❛❝❡ ♦❢ st❛t❡s ♦❢ ❛ ♣r♦t❡✐♥✱ ✇❤✐❝❤ ✐s ❝♦♠♣♦s❡❞ ♦❢ t✇♦ s✉❜st❛t❡s✱ ❡❛❝❤ ♦❢ t❤❡s❡ t✇♦ s✉❜st❛t❡s ✐♥ t✉r♥ ❛r❡ ❝♦♠♣♦s❡❞ ♦❢ t✇♦ s✉❜st❛t❡s ❛♥❞ s♦ ♦♥✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣r♦t❡✐♥ ✐s ♠♦❞❡❧❡❞ ❛s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ tr❡❡ s❤♦✇❡❞ ✐♥ ❋✐❣✉r❡ ✶✳❜✱ ✇❤❡r❡ t❤❡v′_is❛r❡ t❤❡ ❥✉♠♣✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✳ ❋✐❣✉r❡ ✶✳❛ ❣✐✈❡s ❛♥ ❡q✉✐✈❛❧❡♥t✱ ❜✉t ❢r❛❝t❛❧✱ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ ♣r♦t❡✐♥✳ ■♥ t❤❡s❡ ♠♦❞❡❧s t❤❡ st❛t❡s ❛r❡ s❡♣❛r❛t❡❞ ❜② ❡♥❡r❣② ❜❛rr✐❡rs✳

❆✈❡t✐s♦✈ ❡t ❛❧✳ s❤♦✇❡❞ t❤❛t t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣r♦t❡✐♥ ✐s ❝♦♥tr♦❧❧❡❞ ❜② ❛ p−❛❞✐❝

♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠✿

∂f(x, t)

∂t =

Z

Qp

❚❍❊❙■❙ ❖❱❊❘❱■❊❲ ❱

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f(x, t) :Q_p×R₊ →R₊ ✐s ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ t❤❡

❢✉♥❝t✐♦♥ v(x|y) : Q_p ×Q_p → R₊ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ tr❛♥s✐t✐♦♥ ❢r♦♠ st❛t❡ y t♦ t❤❡ st❛t❡ x ♣❡r ✉♥✐t t✐♠❡✳ ❚❤❡r❡ ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❡♥❡r❣② ❧❛♥❞s❝❛♣❡✶ ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ tr❛♥s✐t✐♦♥ ✭❆rr❤❡♥✐✉s r❡❧❛t✐♦♥✮✳ ❇② ❝❤♦♦s✐♥❣ ❝♦♥✈❡♥✐❡♥t❧② t❤❡ ❡♥❡r❣② ❧❛♥❞s❝❛♣❡ ✐♥ t❤✐s r❡❧❛t✐♦♥✱ t❤❡ r✐❣❤t s✐❞❡ ♦❢ ✭✶✮ ❜❡❝♦♠❡s ❱❧❛❞✐♠✐r♦✈ ♦♣❡r❛t♦r✳ ❚❤✉s✱ ✐t ✐s ♥❛t✉r❛❧ t♦ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠✿

      

∂u(x,t)

∂t =κ R

Qn p

u(x−y,t)−u(x,t)

w(y) dny✱ x∈Qnp, t∈R+,

u(x,0) =ϕ∈S Qn_p,

✇❤❡r❡w(y) ❜❡❧♦♥❣s t♦ ❛ ❣❡♥❡r✐❝ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✳

❋✐❣✉r❡ ✶✿ ❍❡r❡v_i′s ❛r❡ r❛t❡s ♦❢ tr❛♥s✐t✐♦♥s✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥|·|_p✱ ❢♦r 1≤i≤3✳

❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ st✉❞② ♦❢ ✉❧tr❛♠❡tr✐❝ ❞✐✛✉s✐♦♥

■♥ ❈❤❛♣t❡r ✷✱ ✇❡ ❝♦♥t✐♥✉❡ ❛♥❞ ❡①t❡♥❞ s♦♠❡ ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts ❣✐✈❡♥ ✐♥ ❬✾❪✱ ❬✶✶❪✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss ♥♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs✿

(W_{ϕ)(x) =κ} Z

Qn p

ϕ(x−y)−ϕ(x)

w|y|_p

dny✱ ❢♦r ϕ∈S(Qn_p),

✇❤❡r❡κ ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❚❤✐s ❝❧❛ss ✐♥❝❧✉❞❡s t❤❡ ❱❧❛❞✐♠✐r♦✈ ♦♣❡r❛t♦r✳ ❚❤❡s❡ ♦♣❡r❛✲ t♦rs ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ r❛❞✐❛❧ ❢✉♥❝t✐♦♥w|y|_p✱ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s t❤❡ str✉❝t✉r❡ ♦❢ t❤❡

❚❍❊❙■❙ ❖❱❊❘❱■❊❲ ❱■

❡♥❡r❣② ❧❛♥❞s❝❛♣❡✳ ❲❡ st✉❞② ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ s♦❧✈❛❜❧❡ ♠♦❞❡❧s✱ ✇❡ ❤❛✈❡ ❝❛❧❧❡❞ t❤❡♠ ♣♦❧②✲ ♥♦♠✐❛❧ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❧❛♥❞s❝❛♣❡s✱ ✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ ❧✐♥❡❛r ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❧❛♥❞s❝❛♣❡s ❝♦♥s✐❞❡r❡❞ ✐♥ ❬✾❪✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✸✳ ❲❡ ❛tt❛❝❤ t♦ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ❛ ▼❛r❦♦✈ ♣r♦❝❡ss✱ ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ❛♥❞ ❤❛s ♥♦ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦t❤❡r t❤❛♥ ❥✉♠♣s✱ s❡❡ ❚❤❡♦r❡♠s ✷✳✶✲✷✳✷✳ ❲❡ ❛❧s♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠✿

  

∂u

∂t(x, t)−Wu(x, t) =g(x, t), x∈Qnp, t∈[0, T], T >0,

u(x,0) =u0(x), u0(x)∈Dom(W),

s❡❡ ❚❤❡♦r❡♠ ✷✳✸✳ ❋✐♥❛❧❧②✱ ✇❡ st✉❞② t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦s ❛tt❛❝❤❡❞ t♦ ♣♦❧②♥♦♠✐❛❧ ❧❛♥❞s❝❛♣❡s✳ ■❢W ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛♥❞♦♠

✇❛❧❦ ✐s r❡❝✉rr❡♥t ✇❤❡♥α≥2n✱ ❛♥❞ ✐t ✐s tr❛♥s✐❡♥t ✇❤❡♥n < α <2n✱ s❡❡ ❚❤❡♦r❡♠ ✷✳✹✳ ❆❧❧ t❤❡ r❡s✉❧ts ❛r❡ ❢♦r♠✉❧❛t❡❞ ✐♥ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥✳ ❚❤❡ r❡s✉❧ts ✐♥ ❈❤❛♣t❡r ✷ ✇❡r❡ ♣✉❜❧✐s❤❡❞ ✐♥ ❬✶✾❪✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡ ✇❛s st✉❞② ✐♥ ❬✶✶❪ ✐♥ ❞✐♠❡♥s✐♦♥ ♦♥❡ ❛♥❞ ✐♥ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥ ✐♥ ❬✶✾❪✱ s❡❡ ❈❤❛♣t❡r ✷✳ ■♥ t❤❡ ❈❤❛♣t❡r ✸✱ ✇❡ ❝♦♥s✐❞❡r ♦♣❡r❛t♦rs ♦✈❡rQ4_p

✇❤♦s❡ s②♠❜♦❧s ❛r❡ ♥♦t r❛❞✐❛❧ ❢✉♥❝t✐♦♥s✳ ❇② ✉s✐♥❣ ❛ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s t♦ t❤♦s❡ ♦❢ ❬✶✶❪ ❛♥❞ ❬✶✾❪✱ ✇❡ st✉❞② t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡ ❢♦r ❛ r❛♥❞♦♠ ✇❛❧❦✱ ✇❤♦s❡ ❞✐str✐❜✉t✐♦♥ ❞❡♥s✐t②Z(x, t)✱x∈Q4_p✱t∈ R₊✱ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✉❧tr❛♠❡tr✐❝ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✿

∂u(x, t)

∂t =−

1 Γ2

p(−α) Z

Q4 p

u(x−y, t)−u(x, t)

|f(y)|α_p+2 d

4_y,

✇❤❡r❡f ✐s ❛♥ ❡❧❧✐♣t✐❝ q✉❛❞r❛t✐❝ ❢♦r♠ ♦❢ ❞✐♠❡♥s✐♦♥ ✹✱ ✇❡ s❤♦✇✱ s❡❡ ❚❤❡♦r❡♠ ✸✳✸✱ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✐s r❡❝✉rr❡♥t ✐❢α ≥2 ❛♥❞ tr❛♥s✐❡♥t ✇❤❡♥ α <2✳ ❇② ✉s✐♥❣ t❤❡

s❛♠❡ t❡❝❤♥✐q✉❡s✱ ✇❡ ♦❜t❛✐♥ s✐♠✐❧❛r r❡s✉❧ts ❢♦r t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡ ♦✈❡r

Q2_p✱ s❡❡ ❚❤❡♦r❡♠ ✸✳✹✳ ❋✐♥❛❧❧②✱ ❜② ✉s✐♥❣ t❤❡ s❛♠❡ t❡❝❤♥✐q✉❡ ♦❢ ❬✶✶❪ ✇❡ ✜♥❞ t❤❡ ❛s②♠♣t♦t✐❝

❜❡❤❛✈✐♦r ❢♦r t❤❡ s✉r✈✐✈❛❧ ♣r♦❜❛❜✐❧✐t②✱ s❡❡ ❚❤❡♦r❡♠ ✸✳✺✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ❈❤❛♣t❡r ✸ ✇❡r❡ ♣✉❜❧✐s❤❡❞ ✐♥ ❬✶✼❪✳

■♥ ❈❤❛♣t❡r ✹✱ ✇❡ st✉❞② ♦♣❡r❛t♦rs ♦❢ t❤❡ ❢♦r♠

(W_{αϕ)(x) =κ} Z

Qn p

ϕ(x−y)−ϕ(x)

wα

kyk_p

dny✱

✇❤❡r❡wα

kyk_p❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ✭✐✮wα

kyk_p✐s ❛ r❛❞✐❛❧✱ ✭✐✐✮wα

kyk_p= 0

✐❢ ❛♥❞ ♦♥❧② ✐❢y= 0 ❛♥❞ ✭✐✐✐✮ t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts C0, C1>0✱ ❛♥❞α > n s✉❝❤ t❤❛t

C0kykα_p ≤wα(kyk_p)≤C1kykα_p ❢♦r ❛♥② y∈Qnp.

❲❡ ❞❡♥♦t❡ ❜② M_λ✱ ✇✐t❤ _{λ ≥ 0}✱ t❤❡ C✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❛❧❧ t❤❡ ❧♦❝❛❧❧② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s

s❛t✐s❢②✐♥❣ |ϕ(x)| ≤ C(1 +kxkλ_p)✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥ ϕ ❞❡♣❡♥❞s ♦♥ ❛ ♣❛r❛♠❡t❡r t✱ ✇❡ s❤❛❧❧ s❛② t❤❛t ϕ❜❡❧♦♥❣s t♦M_λ ✉♥✐❢♦r♠❧② ✇✐t❤ r❡s♣❡❝t t♦ _t✱ ✐❢ ✐ts ❝♦♥st❛♥t_C ❛♥❞ ✐ts ♣❛r❛♠❡t❡r

♦❢ ❧♦❝❛❧ ❝♦♥st❛♥❝② ♦❢ ϕ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t✳ ❚❤❡♥ W_{α :} M_{λ →} M_λ ✐s ✇❡❧❧✲❞❡✜♥❡❞

♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❢♦rα−n > λ✱ s❡❡ ▲❡♠♠❛ ✹✳✶✳

❲❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ❝❧❛ss ♦❢ ♣❛r❛❜♦❧✐❝✲t②♣❡ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ✜①N + 1 ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs s❛t✐s❢②✐♥❣n < α1 < α2 <· · · < αN < α✱

❚❍❊❙■❙ ❖❱❊❘❱■❊❲ ❱■■

♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❲❡ ❛ss✉♠❡ t❤❛t✿ ✭✐✮b(x, t) ❛♥❞ ak(x, t)✱ ❢♦r k= 0, . . . , N✱ ❜❡❧♦♥❣ ✭✇✐t❤

r❡s♣❡❝t t♦ x✮ t♦ M₀ ✉♥✐❢♦r♠❧② ✇✐t❤ r❡s♣❡❝t t♦ _{t ∈[0, T]}❀ ✭✐✐✮ _{a0(x, t)} s❛t✐s✜❡s t❤❡ ❍_¨♦❧❞❡r

❝♦♥❞✐t✐♦♥ ✐♥ t ✇✐t❤ ❡①♣♦♥❡♥t v ∈ (0,1) ✉♥✐❢♦r♠❧② ✐♥ x✳ ❲❡ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤②

♣r♦❜❧❡♠ _

         

∂u

∂t(x, t)−a0(x, t)(Wαu)(x, t)

+PN_k=1ak(x, t)(Wαk)u(x, t) +b(x, t)u(x, t) =f(x, t)

u(x,0) =ϕ(x).

✭✷✮

❈❍❆P❚❊❘

✶

p

−

❛❞✐❝ ❆♥❛❧②s✐s

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✜① t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ ❝♦❧❧❡❝t s♦♠❡ ❜❛s✐❝ r❡s✉❧ts ♦♥ p−❛❞✐❝ ❛♥❛❧②s✐s t❤❛t

✇❡ ✇✐❧❧ ✉s❡ t❤r♦✉❣❤ t❤❡ ❛rt✐❝❧❡✳ ❋♦r ❛ ❞❡t❛✐❧❡❞ ❡①♣♦s✐t✐♦♥ t❤❡ r❡❛❞❡r ♠❛② ❝♦♥s✉❧t ❬✶❪✱ ❬✷✷❪✱ ❬✸✸❪✳

✶✳✶ ❚❤❡ ✜❡❧❞ ♦❢

p

✲❛❞✐❝ ♥✉♠❜❡rs

❆❧♦♥❣ t❤✐s ❛rt✐❝❧❡p✇✐❧❧ ❞❡♥♦t❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r✳ ❚❤❡ ✜❡❧❞ ♦❢p−❛❞✐❝ ♥✉♠❜❡rsQ_p ✐s ❞❡✜♥❡❞

❛s t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡ ✜❡❧❞ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rsQ✇✐t❤ r❡s♣❡❝t t♦ t❤❡p−❛❞✐❝ ♥♦r♠| · |p✱

✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s

|x|p = (

0 ✐❢ x= 0

p−γ _{✐❢ x=p}γa

b,

✇❤❡r❡a ❛♥❞ b❛r❡ ✐♥t❡❣❡rs ❝♦♣r✐♠❡ ✇✐t❤ p✳ ❚❤❡ ✐♥t❡❣❡rγ :=ord(x)✱ ✇✐t❤ ord(0) := +∞✱

✐s ❝❛❧❧❡❞ t❤❡ p−❛❞✐❝ ♦r❞❡r ♦❢ x✳ ❲❡ ❡①t❡♥❞ t❤❡ p−❛❞✐❝ ♥♦r♠ t♦Qn_p ❜② t❛❦✐♥❣

||x||p:= max

1≤i≤n|xi|p, ❢♦rx= (x1, . . . , xn)∈Q n p.

❲❡ ❞❡✜♥❡ ord(x) = min1≤i≤n{ord(xi)}✱ t❤❡♥ ||x||p = p−♦r❞(x)✳ ❚❤❡ s❡t Qnp,|| · ||p ✐s ❛

❝♦♠♣❧❡t❡ ✉❧tr❛♠❡tr✐❝ s♣❛❝❡✳ ❆s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ Q_p ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ ❈❛♥t♦r✲❧✐❦❡

s✉❜s❡t ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✳

❆♥② p−❛❞✐❝ ♥✉♠❜❡r x6= 0 ❤❛s ❛ ✉♥✐q✉❡ ❡①♣❛♥s✐♦♥ x=pord(x)P∞_j=0xjpj✱ ✇❤❡r❡ xj ∈

{0,1,2, . . . , p−1} ❛♥❞ x0 6= 0✳ ❇② ✉s✐♥❣ t❤✐s ❡①♣❛♥s✐♦♥✱ ✇❡ ❞❡✜♥❡ t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢

x∈Q_p✱ ❞❡♥♦t❡❞ {x}p✱ ❛s t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r

{x}p= (

0 ✐❢ x= 0 ♦r ord(x)≥0

p♦r❞(x)P_j−₌₀ord(x)−1xjpj ✐❢ ord(x)<0.

❋♦rγ ∈Z✱ ❞❡♥♦t❡ ❜②B_γn(a) ={x∈Q_pn:||x−a||p≤pγ}t❤❡ ❜❛❧❧ ♦❢ r❛❞✐✉s pγ ✇✐t❤ ❝❡♥t❡r ❛t

a= (a1, . . . , an)∈Qpn✱ ❛♥❞ t❛❦❡Bγn(0) :=Bγn✳ ◆♦t✐❝❡ t❤❛tBγn(a) =Bγ(a1)× · · · ×Bγ(an)✱

✇❤❡r❡ Bγ(ai) := {x ∈ Qp :|x−ai|p ≤pγ} ✐s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❜❛❧❧ ♦❢ r❛❞✐✉s pγ ✇✐t❤

❈❍❆P❚❊❘ ✶✳ P−❆❉■❈ ❆◆❆▲❨❙■❙ ✷

❝❡♥t❡r ❛t ai ∈ Qp✳ ❚❤❡ ❜❛❧❧ B0n ❡q✉❛❧s t❤❡ ♣r♦❞✉❝t ♦❢ n ❝♦♣✐❡s ♦❢ B0 := Zp✱ t❤❡ r✐♥❣

♦❢ p−❛❞✐❝ ✐♥t❡❣❡rs✳ ❲❡ ❞❡♥♦t❡ ❜② Ω(kxk_p) t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ Bn

0✳ ❋♦r ♠♦r❡

❣❡♥❡r❛❧ s❡ts✱ s❛② ❇♦r❡❧ s❡ts✱ ✇❡ ✉s❡1A(x) t♦ ❞❡♥♦t❡ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢A✳

✶✳✷ ❚❤❡ ❇r✉❤❛t✲❙❝❤✇❛rt③ s♣❛❝❡

❆ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ϕ ❞❡✜♥❡❞ ♦♥ Qn_p ✐s ❝❛❧❧❡❞ ❧♦❝❛❧❧② ❝♦♥st❛♥t ✐❢ ❢♦r ❛♥② x ∈ Qn_p

t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡rl=l(x)∈Z s✉❝❤ t❤❛t

ϕ(x+x′) =ϕ(x) ❢♦r x′ ∈Bn_l. ✭✶✳✶✮

❚❤❡ s❡t ♦❢ ❛❧❧ ❧♦❝❛❧❧② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥sϕ✱ ❢♦r ✇❤✐❝❤ t❤❡ ✐♥t❡❣❡rl(x) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢

x✱ ❢♦r♠ C✲✈❡❝t♦r s♣❛❝❡ ❞❡♥♦t❡❞ ❜② Ee(Qn

p) :=Ee✳ ●✐✈❡♥ ϕ∈Ee✱ ✇❡ ❝❛❧❧ t❤❡ ❧❛r❣❡st ♣♦ss✐❜❧❡

l=l(ϕ)✱ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❧♦❝❛❧ ❝♦♥st❛♥❝② ♦❢ ϕ✳

❆ ❢✉♥❝t✐♦♥ ϕ:Qn_p →C✐s ❝❛❧❧❡❞ ❛ ❇r✉❤❛t✲❙❝❤✇❛rt③ ❢✉♥❝t✐♦♥ ✭♦r ❛ t❡st ❢✉♥❝t✐♦♥✮ ✐❢ ✐t ✐s

❧♦❝❛❧❧② ❝♦♥st❛♥t ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt✳ ❚❤❡ C✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❇r✉❤❛t✲❙❝❤✇❛rt③ ❢✉♥❝t✐♦♥s

✐s ❞❡♥♦t❡❞ ❜② S(Qn

p) :=S✳ ◆♦t✐❝❡ t❤❛tS⊂Ee✳

▲❡t S′(Qn_p) :=S′ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥❛❧s ✭❞✐str✐❜✉t✐♦♥s✮ ♦♥ S(Qn_p)✳ ❆❧❧ ❢✉♥❝✲

t✐♦♥❛❧s ♦♥S(Qn

p) ❛r❡ ❝♦♥t✐♥✉♦✉s✳

❙❡t χp(y) = exp(2πi{y}p) ❢♦r y ∈ Qp✳ ❚❤❡ ♠❛♣ χp(·) ✐s ❛♥ ❛❞❞✐t✐✈❡ ❝❤❛r❛❝t❡r ♦♥Qp✱

✐✳❡✳ ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣ ❢r♦♠ Q_p ✐♥t♦ t❤❡ ✉♥✐t ❝✐r❝❧❡ s❛t✐s❢②✐♥❣ χp(y0+y1) = χp(y0)χp(y1)✱

y0, y1 ∈Qp✳

●✐✈❡♥ ξ = (ξ1, . . . , ξn) ❛♥❞ x = (x1, . . . , xn) ∈ Qnp✱ ✇❡ s❡t ξ ·x := Pn

j=1ξjxj✳ ❚❤❡

❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ ϕ∈S(Qn_p) ✐s ❞❡✜♥❡❞ ❛s

(Fϕ)(ξ) =

Z

Qn p

Ψ(−ξ·x)ϕ(x)dnx ❢♦r ξ∈Qn_p,

✇❤❡r❡ dn_{x ✐s t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦♥ Q}n

p ♥♦r♠❛❧✐③❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ vol(B0n) = 1✳ ❚❤❡

❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ S(Qn_p) ♦♥t♦ ✐ts❡❧❢ s❛t✐s❢②✐♥❣(F(Fϕ))(ξ) =

ϕ(−ξ)✳ ❲❡ ✇✐❧❧ ❛❧s♦ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ Fx→ξϕ❛♥❞ ϕb❢♦r t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ϕ✳

✶✳✸ ❋♦✉r✐❡r tr❛♥s❢♦r♠

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠F[T]♦❢ ❛ ❞✐str✐❜✉t✐♦♥ T ∈S′ _Qn p

✐s ❞❡✜♥❡❞ ❜②

(F[T], ϕ) = (T,F[ϕ]) ❢♦r ❛❧❧ϕ∈S Qn_p✳

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ f → F[T] ✐s ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ S′ Qn_p ♦♥t♦ S′ Qn_p✳

❈❍❆P❚❊❘ ✶✳ P−❆❉■❈ ❆◆❆▲❨❙■❙ ✸

✶✳✹ ❊❧❧✐♣t✐❝ Ps❡✉❞♦ ❉✐✛❡r❡♥t✐❛❧ ❖♣❡r❛t♦rs

❉❡✜♥✐t✐♦♥ ✶✳✶✳ ▲❡t f(ξ) ∈ Qn_p[ξ1, . . . , ξn] ❜❡ ❛ ♥♦♥ ❝♦♥st❛♥t ♣♦❧②♥♦♠✐❛❧✳ ❲❡ s❛② t❤❛t

f(ξ) ✐s ❛♥ ❡❧❧✐♣t✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡d, ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s(i) f(ξ) ✐s

❛ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡d, ❛♥❞ (ii) f(ξ) = 0⇔ξ= 0.

❲❡ ♥♦t❡ t❤❛t ✐❢f(ξ)✐s ❡❧❧✐♣t✐❝✱ t❤❡♥cf(ξ) ✐s ❡❧❧✐♣t✐❝ ❢♦r ❛♥②c∈Q×_p.❋♦r t❤✐s r❡❛s♦♥ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ❢r♦♠ ♥♦✇ ♦♥ t❤❛t ❡❧❧✐♣t✐❝ ♣♦❧②♥♦♠✐❛❧s ❤❛✈❡ ❝♦❡✣❝✐❡♥ts ✐♥ Z_p.

❉❡✜♥✐t✐♦♥ ✶✳✷✳ ▲❡tf(ξ)∈Zn_p[ξ1, . . . , ξn]❜❡ ❛ ♥♦♥ ❝♦♥st❛♥t ♣♦❧②♥♦♠✐❛❧✳ ❆ ♣s❡✉❞♦ ❞✐✛❡r✲

❡♥t✐❛❧ ♦♣❡r❛t♦rf(D, α), α >0,✇✐t❤ s②♠❜♦❧ |f(ξ)|α

p✱ ✐s ❛♥ ♦♣❡r❛t♦r ♦❢ t❤❡ ❢♦r♠

(f(D, α)ϕ) :=F_ξ−_→1_x |f|α_pFx→ξϕ, f or ϕ∈S(Qnp).

■❢ f ✐s ❛♥ ❡❧❧✐♣t✐❝ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ s❛✐❞ t❤❛t f(D, α) ✐s ❛♥ ❡❧❧✐♣t✐❝ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧

❈❍❆P❚❊❘

✷

◆♦♥❧♦❝❛❧ ❖♣❡r❛t♦rs✱ P❛r❛❜♦❧✐❝✲t②♣❡ ❊q✉❛t✐♦♥s✱ ❛♥❞

❯❧tr❛♠❡tr✐❝ ❘❛♥❞♦♠ ❲❛❧❦s

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ t②♣❡ ♦❢ ♥♦♥✲❧♦❝❛❧ ♦♣❡r❛t♦rs ❛♥❞ st✉❞② t❤❡ ❈❛✉❝❤② ♣r♦❜✲ ❧❡♠ ❢♦r ❝❡rt❛✐♥ ♣❛r❛❜♦❧✐❝✲t②♣❡ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞ t♦ t❤❡s❡ ♦♣❡r❛t♦rs✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ❛r❡ t❤❡ p−❛❞✐❝ ♠❛st❡r ❡q✉❛t✐♦♥s ♦❢ ❝❡rt❛✐♥ ♠♦❞❡❧s

♦❢ ❝♦♠♣❧❡① s②st❡♠s ✐♥tr♦❞✉❝❡❞ ❜② ❆✈❡t✐s♦✈ ❡t ❛❧✳ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ ♣❛r❛❜♦❧✐❝✲t②♣❡ ❡q✉❛t✐♦♥s ❛r❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ♦♥ t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r t❤❡ ✜❡❧❞ ♦❢ p−❛❞✐❝ ♥✉♠❜❡rs✳ ❲❡ st✉❞② s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ r❛♥❞♦♠

✇❛❧❦s✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ✜rst ♣❛ss❛❣❡ t✐♠❡✳

✷✳✶ ❆ ◆❡✇ ❈❧❛ss ♦❢ ◆♦♥❧♦❝❛❧ ❖♣❡r❛t♦rs

❚❛❦❡ R₊:={x∈R;x≥0}✱ ❛♥❞ ✜① ❛ ❢✉♥❝t✐♦♥

w:Qn_p →R₊

s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✭✐✮ w(y)✐s ❛ r❛❞✐❛❧ ✭✐✳❡✳ w(y) =wkyk_p✮ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥❀

✭✐✐✮w(y) = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢y= 0❀

✭✐✐✐✮ t❤❡r❡ ❡①✐sts ❝♦♥st❛♥tsC0 >0✱M ∈Z✱ ❛♥❞α1> n s✉❝❤ t❤❛t

C0kykαp1 ≤w(kykp)✱ ❢♦r kykp ≥pM.

◆♦t❡ t❤❛t ❝♦♥❞✐t✐♦♥ ✭✐✐✐✮ ✐♠♣❧✐❡s t❤❛t

Z

kyk_p≥pM

dn_y

wkyk_p

<∞. ✭✷✳✶✮

■♥ ❛❞❞✐t✐♦♥✱ s✐♥❝❡w(y) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ ✭✷✳✶✮ ❤♦❧❞s ❢♦r ❛♥②M ∈Z✳ ❈♦♥✈❡r❣❡♥❝❡

❝♦♥❞✐t✐♦♥s ❢♦r ✐♥t❡❣r❛❧ ❦❡r♥❡❧s ♦❢ t②♣❡ ✭✷✳✶✮ ✇❡r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❬✷✻❪✱ ❬✷✼❪ ❛♥❞ ❬✷✸❪✳

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✺

❲❡ ❞❡✜♥❡

(W_{ϕ)(x) =κ} Z

Qn p

ϕ(x−y)−ϕ(x)

w(y) d

n_{y✱ ❢♦r ϕ∈S✱ ✭✷✳✷✮}

✇❤❡r❡κ ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ▲❡♠♠❛ ✷✳✶✳ ❋♦r 1≤ρ≤ ∞✱

S Qn_p → Lρ Qn_p

ϕ → W_ϕ

✐s ❛ ✇❡❧❧✲❞❡✜♥❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❋✉rt❤❡r♠♦r❡✱

F[W_{ϕ] (ξ) =−κ}    Z

Qn p

1−Ψ (−y·ξ)

w(y) d

n_y  

F[ϕ] (ξ). ✭✷✳✸✮

Pr♦♦❢✳ ◆♦t❡ t❤❛t

(W_{ϕ)(x) =κ}

1Qn

prB_pMn (x)

w(x) ∗ϕ(x)−κϕ(x)

  

Z

kykp≥pM

dn_y

w(y)

 

, ✭✷✳✹✮

❢♦r s♦♠❡ ❝♦♥st❛♥t M = M(ϕ)✳ ■❢ ϕ ∈ S ⊂ Lρ✱ ❢♦r 1 ≤ ρ ≤ ∞✱ ✭✷✳✶✮✱ t❤❡♥ t❤❡ ❨♦✉♥❣

✐♥❡q✉❛❧✐t② ✐♠♣❧✐❡s t❤❛t t❤❡ ✜rst t❡r♠ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✷✳✹✮ ❜❡❧♦♥❣s t♦ Lρ _❢♦r

1 ≤ ρ ≤ ∞✱ ❛♥❞ ❜② ✭✷✳✶✮ t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ ✭✷✳✹✮ ❛❧s♦ ❜❡❧♦♥❣s t♦ Lρ ❢♦r 1 ≤ ρ ≤ ∞✳

❋✐♥❛❧❧②✱ ❢♦r♠✉❧❛ ✭✷✳✸✮ ❢♦❧❧♦✇s ❢r♦♠ ❋✉❜✐♥✐✬s t❤❡♦r❡♠✱ s✐♥❝❡

ϕ(x−_wy₍)_y−₎ ϕ(x)

∈L1 Qnp ×Qnp, dnxdny

.

❲❡ s❡t

Aw(ξ) := Z

Qn p

1−Ψ (−y·ξ)

w(y) d

n_y.

▲❡♠♠❛ ✷✳✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ Aw(ξ) ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❢♦r kξkp = p−γ 6= 0✱

✇✐t❤ γ =ord(ξ)✱

Aw p−γ= (1−p−n)

∞

X

j=γ+2

pnj

w(pj₎ +

pnγ

w(pγ+1₎; ✭✷✳✺✮

✭✐✐✮ ✐t ✐s r❛❞✐❛❧✱ ♣♦s✐t✐✈❡✱ ❝♦♥t✐♥✉♦✉s✱ ❛♥❞ Aw(0) = 0✱ ✭✐✐✐✮ Aw p−ord(ξ) ✐s ❛ ❞❡❝r❡❛s✐♥❣

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✻

Pr♦♦❢✳ ❲❡ ✇r✐t❡ ξ=pγξ0,✇✐t❤γ =ord(ξ)❛♥❞ kξ0kp= 1✳ ❚❤❡♥

Aw(ξ) = Z

Qn p

1−Ψ (−pγy·ξ0)

wkyk_p

dny=pγn

Z

Qn p

1−Ψ (−z·ξ0)

wpγ_kzk p

dnz. ✭✷✳✻✮

❲❡ ♥♦✇ ♥♦t❡ t❤❛t

Qn_p r{0}= G

j∈Z

pjU

✇✐t❤

U :=ny ∈Qn_p;kyk_p = 1o. ❇② ✉s✐♥❣ t❤✐s ♣❛rt✐t✐♦♥ ❛♥❞ ✭✷✳✻✮✱ ✇❡ ❤❛✈❡

Aw(ξ) = X

j∈Z

pγn

Z

pj_U

1−Ψ (−z·ξ0)

wpγ_kzk p

dnz

=X

j∈Z

p−jn+γn

w(p−j+γ₎    1−p

−n₋Z

U

Ψ −pjy·ξ0dny

  .

❇② ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛

Z

U

Ψ −pjy·ξ0dny=

          

1−p−n ✐❢ j≥0

−p−n _{✐❢ j=−1}

0 ✐❢ j <−1,

✭✷✳✼✮

✇❡ ❣❡t

Aw(ξ) = (1−p−n)

∞

X

j=2

pn(γ+j)

w(pγ+j₎ +

pnγ

w(pγ+1₎

= (1−p−n)

∞

X

j=γ+2

pnj

w(pj₎+

pnγ

w(pγ+1₎. ✭✷✳✽✮

❋r♦♠ ✭✷✳✽✮ ❢♦❧❧♦✇s t❤❛t Aw(ξ) ✐s r❛❞✐❛❧✱ ♣♦s✐t✐✈❡✱ ❝♦♥t✐♥✉♦✉s ♦✉ts✐❞❡ ♦❢ t❤❡ ♦r✐❣✐♥✱ ❛♥❞

t❤❛tAw p−ord(ξ) ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ord(ξ)✳ ❚♦ s❤♦✇ t❤❡ ❝♦♥t✐♥✉✐t② ❛t ♦r✐❣✐♥✱ ✇❡

♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✳ ❙✐♥❝❡P∞

j=M p nj

w(pj₎ <∞✱ ❝✳❢✳ ✭✷✳✶✮✱

Aw(0) := lim γ→∞(1−p

−n₎

∞

X

j=γ+2

pnj

w(pj₎ + lim_γ→∞

pnγ

w(pγ+1₎ = 0.

Pr♦♣♦s✐t✐♦♥ ✷✳✶✳ ✭✐✮ (W_{ϕ) (x) = −κF}−1 ξ→x

Aw(kξk_p)Fx→ξϕ

❢♦r ϕ ∈ S Qn_p✱ ❛♥❞

Wϕ ∈ C Qn_p ∩Lρ _Qn p

✱ ❢♦r 1 ≤ ρ ≤ ∞✳ ❚❤❡ ❖♣❡r❛t♦r W ❡①t❡♥❞s t♦ ❛♥ ✉♥❜♦✉♥❞❡❞

❛♥❞ ❞❡♥s❡❧② ❞❡✜♥❡❞ ♦♣❡r❛t♦r ✐♥ L2 Qn_p ✇✐t❤ ❞♦♠❛✐♥

Dom(W_{) =} n

ϕ∈L2;Aw(kξkp)Fϕ∈L2 o

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✼

✭✐✐✮ (−W_{, Dom(}W₎₎ ✐s s❡❧❢✲❛❞❥♦✐♥t ❛♥❞ ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r✳

✭✐✐✐✮ W ✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ ❛ ❝♦♥tr❛❝t✐♦♥ _C0 s❡♠✐❣r♦✉♣ _(T(t))

t≥0✳ ▼♦r❡♦✈❡r✱

t❤❡ s❡♠✐❣r♦✉♣ (T(t))_t≥0 ✐s ❜♦✉♥❞❡❞ ❤♦❧♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥❣❧❡ π/2✳

Pr♦♦❢✳ ✭✐✮ ■t ❢♦❧❧♦✇s ❢r♦♠ ▲❡♠♠❛ ✷✳✶ ❛♥❞ t❤❡ ❢❛❝t t❤❛tAw(kξk_p)✐s ❝♦♥t✐♥✉♦✉s✱ ❝✳❢✳ ▲❡♠♠❛

✷✳✷✳ ✭✐✐✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛tW ✐s ❛ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❛♥❞ t❤❛t t❤❡ ❋♦✉r✐❡r

tr❛♥s❢♦r♠ ♣r❡s❡r✈❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢L2_{✳ ✭✐✐✐✮ ■t ❢♦❧❧♦✇s ♦❢ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✱ s❡❡ ❡✳❣✳}

❬✷✶✱ ❈❤❛♣✳ ✷✱ ❙❡❝t✳ ✸❪ ♦r ❬✶✻❪✳ ❋♦r t❤❡ ♣r♦♣❡rt② ♦❢ t❤❡ s❡♠✐❣r♦✉♣ ♦❢ ❜❡✐♥❣ ❤♦❧♦♠♦r♣❤✐❝✱ s❡❡ ❡✳❣✳ ❬✷✶✱ ❈❤❛♣✳ ✷✱ ❙❡❝t✳ ✹✳✼❪✳

✷✳✷ ❙♦♠❡ ❛❞❞✐t✐♦♥❛❧ r❡s✉❧ts

▲❡♠♠❛ ✷✳✸✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts α1✱α2✱ C0✱ C1✱ ✇✐t❤ α1 > n✱

α2> n✱ ❛♥❞ α3 ≥0✱ s✉❝❤ t❤❛t

C0

ξ′α1

p ≤w(

ξ′_p)≤C1

ξ′α2

p e

α3kξ′kp✱ ❢♦r ❛♥② _ξ′ _∈Qn

p. ✭✷✳✶✵✮

❚❤❡♥ t❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ ❝♦♥st❛♥tsC2✱ C3✱ s✉❝❤ t❤❛t

C2kξkpα2−ne−α3pkξk

−1 p ≤A

w(kξkp)≤C3kξkαp1−n

❢♦r ❛♥② ξ ∈Qn_p✱ ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t e−α3pk0k−p1 _{:= lim}

kξk_p→0e−α3pkξk

−1

p _{= 0}✳ ❋✉rt❤❡r✲

♠♦r❡✱ ✐❢ α3 >0✱ t❤❡♥ α1≥α2✱ ❛♥❞ ✐❢ α3= 0✱ t❤❡♥ α1 =α2✳

Pr♦♦❢✳ ❇② ✉s✐♥❣ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r w❣✐✈❡♥ ✐♥ ✭✷✳✶✵✮✱ ❛♥❞ kξk_p =p−γ_✱

Aw(kξk_p)≤ (1−p

−n₎

C0

∞

X

j=γ+2

pnj pjα1 +

pnγ

pα1(γ+1) ≤C3kξk α1−n p .

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Aw

kξk_p≥ _w(p_pnγγ+1₎ ✱ ❛♥❞ ❜② ✉s✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r w ❣✐✈❡♥

✐♥ ✭✷✳✶✵✮✱

Aw

kξk_p≥ p

nγ

w(pγ+1₎ ≥

pnγ

C1pα2(γ+1)eα3pγ+1

≥C2kξk_pα2−ne−α3pkξk

−1 p .

❉❡✜♥✐t✐♦♥ ✷✳✶✳ ❲❡ s❛② t❤❛tW ✭♦r_Aw✮ ✐s ♦❢ ❡①♣♦♥❡♥t✐❛❧ t②♣❡ ✐❢ ✐♥❡q✉❛❧✐t② ✭✷✳✶✵✮ ✐s ♦♥❧②

♣♦ss✐❜❧❡ ❢♦r α3 > 0 ✇✐t❤α1✱α2✱C0✱ C1 ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ❛♥❞α1 > n✱ α2 > n✳ ■❢ ✭✷✳✶✵✮

❤♦❧❞s ❢♦r α3 = 0 ✇✐t❤ α1✱α2✱ C0✱ C1 ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ❛♥❞ α1 > n✱ α2 > n✱ ✇❡ s❛② t❤❛t

W ✭♦r_Aw✮ ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡✳

❲❡ ♥♦t❡ t❤❛t ✐❢ W ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡ t❤❡♥ _{α1 = α2 > n} ❛♥❞ _C0✱ _C1 ❛r❡ ♣♦s✐t✐✈❡

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✽

▲❡♠♠❛ ✷✳✹✳ ❲✐t❤ t❤❡ ❤②♣♦t❤❡s❡s ♦❢ ▲❡♠♠❛ ✷✳✸✱

e−tκAw(kξkp)_∈Lρ_(Qn

p) ❢♦r 1≤ρ <∞ ❛♥❞ t >0.

Pr♦♦❢✳ ❙✐♥❝❡ e−tAw(kξkp) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✱ ✐t ✐s s✉✣❝✐❡♥t t♦ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts

M ∈Ns✉❝❤ t❤❛t

IM(t) := Z

kξkp>pM

e−ρκtAw(kξkp)_dn_{ξ <∞,} ❢♦r _{t >0}✳

❚❛❦❡M ∈N✱ ❜② ▲❡♠♠❛ ✷✳✸✱ ✇❡ ❤❛✈❡

C2kξkpα2−ne−α3pkξk

−1 p > C

2kξkαp2−ne−α3p

−M+1

❢♦r kξk_p > pM,

❛♥❞ ✭✇✐t❤B =C2κe−α3p

−M+1

✮✱

IM(t)≤ Z

kξk_p>pM

e−tBkξkpα2−n_dn_{ξ ≤C(M, κ, ρ)t}

−n

α₂−n_, ❢♦r _{t >0}✳

✷✳✸

p

✲❛❞✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ r❡❧❛t✐♦♥ ✐♥ ❝♦♠♣❧❡①

s②st❡♠s

■♥ ❬✾❪ ❆✈❡t✐s♦✈ ❡t ❛❧✳ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ r❡❧❛①❛t✐♦♥ ♣r♦❝❡ss❡s ✐♥ ❝♦♠♣❧❡① s②st❡♠s ✭s✉❝❤ ❛s ❣❧❛ss❡s✱ ♠❛❝r♦♠♦❧❡❝✉❧❡s ❛♥❞ ♣r♦t❡✐♥s✮ ♦♥ t❤❡ ❜❛s✐s ♦❢ p✲❛❞✐❝ ❛♥❛❧②s✐s✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ ❝♦♠♣❧❡① s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ r❛♥❞♦♠ ✇❛❧❦ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥✜❣✉r❛t✐♦♥❛❧ st❛t❡s✱ ✇❤✐❝❤ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛♥ ✉❧tr❛♠❡tr✐❝ s♣❛❝❡ ✭Q_p✮✳ ▼❛t❤❡♠❛t✲

✐❝❛❧❧② s♣❡❛❦✐♥❣✱ t❤❡ t✐♠❡✲ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s ❝♦♥tr♦❧❧❡❞ ❜② ❛ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠

∂f(x, t)

∂t =

Z

Qp

{v(x|y)f(y, t)−v(y |x)f(x, t)}dy✱x∈Q_p✱t∈R₊, ✭✷✳✶✶✮

✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ f(x, t) :Q_p×R₊ →R₊ ✐s ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ t❤❡

❢✉♥❝t✐♦♥ v(x|y) :Q_p×Q_p→R₊ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ tr❛♥s✐t✐♦♥ ❢r♦♠ st❛t❡y t♦ t❤❡ st❛t❡ x♣❡r ✉♥✐t t✐♠❡✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ❛ st❛t❡yt♦ ❛ st❛t❡x❝❛♥ ❜❡ ♣❡r❝❡✐✈❡❞ ❛s ♦✈❡r❝♦♠✐♥❣ t❤❡ ❡♥❡r❣② ❜❛rr✐❡r s❡♣❛r❛t✐♥❣ t❤❡s❡ st❛t❡s✳ ■♥ ❬✾❪ ❛♥ ❆rr❤❡♥✐✉s t②♣❡ r❡❧❛t✐♦♥ ✇❛s ✉s❡❞✿

v(x|y)∼A(T) exp

−U(x|y)

kT

,

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✾

v(x|y)✳ ❚❤❡ ❝❛s❡ v(x|y) = v(y|x) ❝♦rr❡s♣♦♥❞s t♦ ❛ ❞❡❣❡♥❡r❛t❡ ❡♥❡r❣② ❧❛♥❞s❝❛♣❡✳ ■♥

t❤✐s ❝❛s❡ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ✭✷✳✶✶✮ t❛❦❡s t❤❡ ❢♦r♠ ∂f(x, t)

∂t =

Z

Qp

v|x−y|_p{f(y, t)−f(x, t)}dy✱

✇❤❡r❡v|x−y|_p= _|xA₋(T_y|)

pexp

−U(|x−y|p) kT

✳ ❇② ❝❤♦♦s✐♥❣U ❝♦♥✈❡♥✐❡♥t❧②✱ s❡✈❡r❛❧ ❡♥❡r❣② ❧❛♥❞s❝❛♣❡s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳ ❋♦❧❧♦✇✐♥❣ ❬✾❪✱ t❤❡r❡ ❛r❡ t❤r❡❡ ❜❛s✐❝ ❧❛♥❞s❝❛♣❡s✿ ✭✐✮ ✭❧♦❣❛r✐t❤✲ ♠✐❝✮ v|x−y|_p= 1

|x−y|plnα(1+|x−y|p)✱

α >1 ✭✐✐✮ ✭❧✐♥❡❛r✮ v|x−y|_p= 1

|x−y|αp+1✱

α >0✱

✭✐✐✐✮ ✭❡①♣♦♥❡♥t✐❛❧✮ v|x−y|_p= e−_|xα₋|x_y−_|y|p

p ✱α >0✳

❚❤✉s✱ ✐t ✐s ♥❛t✉r❛❧ t♦ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ ❈❛✉❝❤② ♣r♦❜❧❡♠✿

      

∂u(x,t)

∂t =κ R

Qn p

u(x−y,t)−u(x,t)

w(y) dny✱ x∈Qnp, t∈R+,

u(x,0) =ϕ∈S Qn_p,

✇❤❡r❡ w(y) ✐s ❛ r❛❞✐❛❧ ❢✉♥❝t✐♦♥ ❜❡❧♦♥❣✐♥❣ t♦ ❛ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❝♦♥t❛✐♥s ❢✉♥❝t✐♦♥s

❧✐❦❡✿

✭✐✮ w(kyk_p) = Γn_p(−α)kykα_p+n✱ ❤❡r❡ Γn_p(·) ✐s t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ p✲❛❞✐❝ ●❛♠♠❛ ❢✉♥❝t✐♦♥✱

❛♥❞ α >0❀

✭✐✐✮w(kyk_p) =kykβ_peαkykp✱_{α >0}✳

❲❡ r❡❝❛❧❧ t❤❛t ♦♣❡r❛t♦r W ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❝❛s❡ ✭✐✮ ✐s t❤❡ ❚❛✐❜❧❡s♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤

✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❱❧❛❞✐♠✐r♦✈ ♦♣❡r❛t♦r✱ s❡❡ ❬✷✽❪✳

❇② ✐♠♣♦s✐♥❣ ❝♦♥❞✐t✐♦♥ ✭✷✳✶✵✮ t♦ w✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ ❜❛s✐❝ ❡♥❡r❣✐❡s ❧❛♥❞s❝❛♣❡s ✐♥ ♦✉r st✉❞②✳ ❚❛❦❡w(kyk_p) s❛t✐s❢②✐♥❣ ✭✷✳✶✵✮ ❛♥❞ t❛❦❡fkyk_p❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t

0< sup

y∈Qn p

fkyk_p<∞ ❛♥❞0< inf

y∈Qn p

fkyk_p<∞.

❚❤❡♥ fkyk_pw(kyk_p) s❛t✐s✜❡s ✭✷✳✶✵✮✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❛❦❡ P(kyk_p) t♦ ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ kyk_p ✇✐t❤ r❡❛❧ ♣♦s✐t✐✈❡ ❝♦❡✣✲

❝✐❡♥ts ❛♥❞ ♥♦♥③❡r♦ ❝♦♥st❛♥t t❡r♠✱ t❤✉sinfy∈Qn pP

kyk_p=P(0)>0✱ ❛♥❞ t❛❦❡w(kyk_p) =

kykβ_peαkykp s❛t✐s❢②✐♥❣ ✭✷✳✶✵✮✱ t❤❡♥_P

kyk_pw(kyk_p) ❛❧s♦ s❛t✐s✜❡s ✭✷✳✶✵✮✳

❋✐♥❛❧❧② ✇❡ ♥♦t❡ t❤❛t kykβ_plnα(1 +kyk_p)✱ β > n✱ α ∈ N✱ ❞♦❡s ♥♦t s❛t✐s✜❡s kykα1 p ≤

kykβ_plnα(1 +kyk_p) ❢♦r ❛♥② y ∈ Qn_p✱ ❛♥❞ ❤❡♥❝❡ ♦✉r r❡s✉❧ts ❞♦ ♥♦t ✐♥❝❧✉❞❡ t❤❡ ❝❛s❡ ♦❢

❧♦❣❛r✐t❤♠✐❝ ❧❛♥❞s❝❛♣❡s✳

✷✳✹ ❍❡❛t ❑❡r♥❡❧s

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✶✵

❲❡ ❞❡✜♥❡

Z(x, t;w, κ) :=Z(x, t) =

Z

Qn p

e−κtAw(kξkp)Ψ(x·ξ)dnξ ❢♦r t >0❛♥❞ x∈_Qn p✳

◆♦t❡ t❤❛t ❜② ▲❡♠♠❛ ✷✳✹✱ Z(x, t) =F_ξ−_→1_x[e−κtAw(kξkp)_]∈L1_∩L2 ❢♦r _{t >0}✳ ❲❡ ❝❛❧❧ ❛ s✉❝❤

❢✉♥❝t✐♦♥ ❛ ❤❡❛t ❦❡r♥❡❧✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ Z(x, t) ❛s ❛ ❢✉♥❝t✐♦♥

♦❢ x ❢♦r t✜①❡❞ ✇❡ ✇✐❧❧ ✇r✐t❡ Zt(x)✳

▲❡♠♠❛ ✷✳✺✳ ❚❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C✱ s✉❝❤ t❤❛t

Z(x, t)< Ctkxk−α1

p ✱ ❢♦r x∈Qnp r{0} ❛♥❞ t >0✳

Pr♦♦❢✳ ▲❡t kxk_p =pβ_{✳ ❙✐♥❝❡ Z(x, t) ∈L}1_(Qn

p) ❢♦r t >0✱ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡

❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ❣❡t

Z(x, t) =kxk−_pn



(1−p−n)

∞

X

j=0

e−κAw(p−β−j)tp−nj−e−κAw(p−β+1)t

 .

❇② ✉s✐♥❣ t❤❛te−κAw(p−β−j)t≤1 ❢♦r j∈N✱ ✇❡ ❤❛✈❡

Z(x, t)≤ kxk−_pnh1−e−κAw(p−β+1)ti.

❲❡ ♥♦✇ ❛♣♣❧② t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ t♦ t❤❡ r❡❛❧ ❢✉♥❝t✐♦♥ f(u) =e−κAw(p−β+1_)u

♦♥

[0, t]✇✐t❤t >0✱ ❛♥❞ ▲❡♠♠❛ ✷✳✸✱

Z(x, t)≤C0kxk−_pntAw(p−β+1)≤Ctkxk−_pα1.

▲❡♠♠❛ ✷✳✻✳ Z(x, t)≥0✱ ❢♦r x∈Qn_p ❛♥❞t >0✳

Pr♦♦❢✳ ❙✐♥❝❡e−tAw(kξkp) ✐s ✐♥t❡❣r❛❜❧❡ ❢♦r _{t >0}❛♥❞ r❛❞✐❛❧✱ ✇❡ ❤❛✈❡

Z(x, t) =

∞

X

i=−∞

e−tAw(pi)

Z

kξkp=pi

Ψ(x·ξ)dnξ

=

∞

X

i=−∞

pnihe−κtAw(pi)−e−κtAw(pi+1)iΩ(p−ix_p)≥0

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✶✶

❚❤❡♦r❡♠ ✷✳✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ Z(x, t) ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✭✐✮ Z(x, t)≥0 ❢♦r ❛♥② t >0❀

✭✐✐✮R

Qn p

Z(x, t)dnx= 1 ❢♦r ❛♥② t >0❀

✭✐✐✐✮ Zt(x)∈C(Qnp,R)∩L1(Qnp)∩L2(Qnp) ❢♦r ❛♥② t >0❀

✭✐✈✮ Zt(x)∗Zt′(x) =Z_t+t′(x) ❢♦r ❛♥② t✱ t′>0❀

✭✈✮ lim

t→0+Z(x, t) =δ(x) ✐♥ S

′_(Qn p)✳

Pr♦♦❢✳ ✭✐✮ ■t ❢♦❧❧♦✇s ❢r♦♠ ▲❡♠♠❛ ✷✳✻✳ ✭✐✐✮ ❋♦r ❛♥② t > 0 t❤❡ ❢✉♥❝t✐♦♥ e−κtAw(kξkp) ✐s

❝♦♥t✐♥✉♦✉s ❛t ξ = 0 ❛♥❞ ❜② ▲❡♠♠❛ ✷✳✹ ✇❡ ❤❛✈❡ e−κtAw(kξkp) _{∈ L}1_∩L2 ❢♦r _{t > 0}✱ t❤❡♥

Zt(x) ∈L1∩L2 ❢♦r t >0✳ ◆♦✇ t❤❡ st❛t❡♠❡♥t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✐♥✈❡rs✐♦♥ ❢♦r♠✉❧❛ ❢♦r t❤❡

❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ✭✐✐✐✮ ❋r♦♠ ▲❡♠♠❛ ✷✳✹✱ ✇✐t❤ ρ = 1,2✱ ✇❡ ❤❛✈❡ Zt(x) ∈ C(Qnp,R)∩

L1(Qn_p)✱ t > 0✱ ❛♥❞ ❜② ✭✐✮ ❛♥❞ ✭✐✐✮✱ Zt(x) ∈ L2(Qnp)✳ ✭✐✈✮ ❇② t❤❡ ♣r❡✈✐♦✉s ♣r♦♣❡rt②

Zt(x)∈L1 ❢♦r ❛♥② t >0✱ t❤❡♥

Zt(x)∗Zt′(x) =F−1

ξ→x

e−κtAw(kξkp)_e−κt′Aw(kξkp)

=F_ξ−_→1_xe−κ(t+t′)Aw(kξkp)

=Zt+t′(x).

✭✈✮ ❙✐♥❝❡ ✇❡ ❤❛✈❡ e−κtAw(kξkp) _{∈ C(Q}n

p,R)∩L1 ❢♦r t > 0✱ ❝✳❢✳ ▲❡♠♠❛ ✷✳✹✱ t❤❡ ✐♥♥❡r

♣r♦❞✉❝t _D

e−κtAw(kξkp)_{, φ} E

=

Z

Qn p

e−κtAw(kξkp)_φ(ξ)dn_ξ

❞❡✜♥❡s ❛ ❞✐str✐❜✉t✐♦♥ ♦♥Qn_p✱ t❤❡♥✱ ❜② t❤❡ ❉♦♠✐♥❛t❡❞ ❈♦♥✈❡r❣❡ ❚❤❡♦r❡♠✱

lim

t→0+ D

e−κtAw(kξkp)_{, φ} E

=h1, φi

❛♥❞ t❤✉s

lim

t→0+hZ(x, t), φi= lim_t→0+ D

e−κtAw(kξkp),F−1φ E

=1,F−1φ= (δ, φ).

✷✳✺ ▼❛r❦♦✈ Pr♦❝❡ss❡s ♦✈❡r

Q

❆❧♦♥❣ t❤✐s s❡❝t✐♦♥ ✇❡ ❝♦♥s✐❞❡rQn_p,k·k_p❛s ❝♦♠♣❧❡t❡ ♥♦♥✲❆r❝❤✐♠❡❞❡❛♥ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞

✉s❡ t❤❡ t❡r♠✐♥♦❧♦❣② ❛♥❞ r❡s✉❧ts ♦❢ ❬✶✸✱ ❈❤❛♣t❡rs ❚✇♦✱ ❚❤r❡❡❪✳ ▲❡t B ❞❡♥♦t❡ t❤❡ ❇♦r❡❧

σ−❛❧❣❡❜r❛ ♦❢Qn_p✳ ❚❤✉s Qn_p,B, dn_{x ✐s ❛ ♠❡❛s✉r❡ s♣❛❝❡✳}

❲❡ s❡t

❈❍❆P❚❊❘ ✷✳ P❆❘❆❇❖▲■❈✲❚❨P❊ ❊◗❯❆❚■❖◆❙ ❆◆❉ ❯▲❚❘❆▼❊❚❘■❈ ❘❆◆❉❖▼ ❲❆▲❑❙ ✶✷

❛♥❞

P(t, x, B) =

(R

Bp(t, y, x)dny ❢♦r t >0, x∈Qnp, B ∈ B

1_B(x) ❢♦r _{t= 0.}

▲❡♠♠❛ ✷✳✼✳ ❲✐t❤ t❤❡ ❛❜♦✈❡ ♥♦t❛t✐♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❤♦❧❞✿ ✭✐✮ p(t, x, y) ✐s ❛ ♥♦r♠❛❧ tr❛♥s✐t✐♦♥ ❞❡♥s✐t②❀

✭✐✐✮ P(t, x, B) ✐s ❛ ♥♦r♠❛❧ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✳

Pr♦♦❢✳ ❚❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ ❚❤❡♦r❡♠ ✷✳✶✱ s❡❡ ❬✶✸✱ ❙❡❝t✐♦♥ ✷✳✶❪ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s✳ ▲❡♠♠❛ ✷✳✽✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ P(t, x, B) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s✿

✭✐✮ ❢♦r ❡❛❝❤ u≥0 ❛♥❞ ❝♦♠♣❛❝t B

lim

x→∞ _tsup_≤u P(t, x, B) = 0 ❬❈♦♥❞✐t✐♦♥ ▲✭❇✮❪❀

✭✐✐✮ ❢♦r ❡❛❝❤ ǫ >0 ❛♥❞ ❝♦♠♣❛❝t B

lim

t→0+sup x∈B

P(t, x,Qn_p \B_ǫn(x)) = 0 ❬❈♦♥❞✐t✐♦♥ ▼✭❇✮❪✳

Pr♦♦❢✳ ✭✐✮ ❇② ▲❡♠♠❛ ✷✳✺ ❛♥❞ t❤❡ ❢❛❝t t❤❛tk·k_p ✐s ❛♥ ✉❧tr❛♥♦r♠✱ ✇❡ ❤❛✈❡

P(t, x, B)≤Ct

Z

B

kx−yk−α1

p dny =tCkxk

−α1

p vol(B) ❢♦r x∈Qnp \B.

❚❤❡r❡❢♦r❡ lim

x→∞sup_t≤uP(t, x, B) = 0✳

✭✐✐✮ ❆❣❛✐♥✱ ❜② ▲❡♠♠❛ ✷✳✺✱ t❤❡ ❢❛❝t t❤❛t k·k_p ✐s ❛♥ ✉❧tr❛♥♦r♠✱ ❛♥❞α1 > n✱ ✇❡ ❤❛✈❡

P(t, x,Qn_p \B_ǫn(x))≤Ct

Z

kx−ykp>ǫ

kx−yk−α1

p dny=Ct Z

kzkp>ǫ

kzk−α1 p dnz

=C′(α1, ǫ, n)t.

❚❤❡r❡❢♦r❡

lim

t→0+sup x∈B

P(t, x,Qn_p \B_ǫn(x))≤ lim

t→0+sup x∈B

C′(α1, ǫ, n)t= 0.

❚❤❡♦r❡♠ ✷✳✷✳ Z(x, t) ✐s t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t② ♦❢ ❛ t✐♠❡ ❛♥❞ s♣❛❝❡ ❤♦♠♦❣❡♥❡♦✉s ▼❛r❦♦✈

♣r♦❝❡ss ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞✱ r✐❣❤t✲❝♦♥t✐♥✉♦✉s ❛♥❞ ❤❛s ♥♦ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦t❤❡r t❤❛♥ ❥✉♠♣s✳

Pr♦♦❢✳ ❚❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ ❬✶✸✱ ❚❤❡♦r❡♠ ✸✳✻❪ ❜② ✉s✐♥❣ t❤❛t(Qn_p,kxk_p)✐s s❡♠✐✲❝♦♠♣❛❝t

s♣❛❝❡✱ ✐✳❡✳ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡ ❜❛s❡✱ ❛♥❞ P(t, x, B) ✐s ❛