Schauder estimates for non-local operators Franziska K uhn - - PowerPoint PPT Presentation

Introduction Definitions Results Outlook

Schauder estimates for non-local operators Franziska K¨ uhn (Institut de Math´ ematiques de Toulouse) Probability and Analysis 2019

May 23, 2019

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Outline

1 Introduction 2 Definitions 3 Schauder estimates for L´

evy operators

4 Outlook: Schauder estimates for L´

evy-type operators

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Introduction

Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Introduction

Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process). Questions: How regular is f ∈ D(A)? If g has a certain regularity then what additional information do we get on the regularity of f ?

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Introduction

Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process). Questions: How regular is f ∈ D(A)? If g has a certain regularity then what additional information do we get on the regularity of f ? Formally, f = A−1g i.e. we are interested in the smoothing properties of A−1.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example I

Laplacian: Af (x) = 1 2∆f (x) . . . appears as infinitesimal generator of Brownian motion.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example I

Laplacian: Af (x) = 1 2∆f (x) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D(A) then f is “almost twice differentiable”: ∥f ∥C2

b(Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞). Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example I

Laplacian: Af (x) = 1 2∆f (x) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D(A) then f is “almost twice differentiable”: ∥f ∥C2

b(Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞).

C k

b (Rd) ⊂ Ck b(Rd) for k ∈ N and C α b (Rd) = Cα b(Rd) for α ∉ N

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example I

Laplacian: Af (x) = 1 2∆f (x) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D(A) then f is “almost twice differentiable”: ∥f ∥C2

b(Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞).

If Af = g ∈ Cκ

b(Rd) for some κ > 0 then

∥f ∥C2+κ

b

(Rd) ≤ Mκ(∥f ∥∞ + ∥Af ∥Cκ

b (Rd)).

C k

b (Rd) ⊂ Ck b(Rd) for k ∈ N and C α b (Rd) = Cα b(Rd) for α ∉ N

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Fractional Laplacian: Af (x) = −(−∆)α/2f (x) ∶= c ∫y≠0 (f (x + y) − f (x) − ∇f (x)y1(0,1)(∣y∣)) 1 ∣y∣d+α dy . . . appears as infinitesimal generator of isotropic α-stable L´ evy process.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Fractional Laplacian: Af (x) = −(−∆)α/2f (x) ∶= c ∫y≠0 (f (x + y) − f (x) − ∇f (x)y1(0,1)(∣y∣)) 1 ∣y∣d+α dy . . . appears as infinitesimal generator of isotropic α-stable L´ evy process. Known: If f ∈ D(A) then ∥f ∥Cα

b (Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞). Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Fractional Laplacian: Af (x) = −(−∆)α/2f (x) ∶= c ∫y≠0 (f (x + y) − f (x) − ∇f (x)y1(0,1)(∣y∣)) 1 ∣y∣d+α dy . . . appears as infinitesimal generator of isotropic α-stable L´ evy process. Known: If f ∈ D(A) then ∥f ∥Cα

b (Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞).

If Af = g ∈ Cκ

b(Rd) for some κ > 0 then

∥f ∥Cα+κ

b

(Rd) ≤ Mκ(∥f ∥∞ + ∥Af ∥Cκ

b (Rd)). Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Infinitesimal generator

. . . appears in the study of Markov processes. Idea: Af = d dt Ptf ∣

t=0

= lim

t→0

Ptf − f t where Ptf (x) ∶= Exf (Xt)

L´ evy

= E0f (x + Xt). is the semigroup.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Infinitesimal generator

. . . appears in the study of Markov processes. Idea: Af = d dt Ptf ∣

t=0

= lim

t→0

Ptf − f t where Ptf (x) ∶= Exf (Xt)

L´ evy

= E0f (x + Xt). is the semigroup. Hence, Exf (Xt) ≈ f (x) + tAf (x) for small t i.e. A describes small-time asymptotics.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Generator of a L´ evy process

Theorem Let (Xt)t≥0 be a L´ evy process. If f ∈ C ∞

c (Rd) then

Af (x) = b ⋅ ∇f (x) + 1 2 tr(Q ⋅ ∇2f (x)) + ∫y≠0(f (x + y) − f (x) − ∇f (x) ⋅ y1(0,1)(∣y∣))ν(dy)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Generator of a L´ evy process

Theorem Let (Xt)t≥0 be a L´ evy process. If f ∈ C ∞

c (Rd) then

Af (x) = b ⋅ ∇f (x) + 1 2 tr(Q ⋅ ∇2f (x)) + ∫y≠0(f (x + y) − f (x) − ∇f (x) ⋅ y1(0,1)(∣y∣))ν(dy) where the L´ evy triplet (b,Q,ν) consists of b ∈ Rd (drift vector), Q ∈ Rd×d symmetric and positive semidefinite (diffusion matrix) a measure ν with ∫y≠0 min{1,∣y∣2}ν(dy) < ∞ (L´ evy measure)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Generator of a L´ evy process

Theorem Let (Xt)t≥0 be a L´ evy process. If f ∈ C ∞

c (Rd) then

Af (x) = b ⋅ ∇f (x) + 1 2 tr(Q ⋅ ∇2f (x)) + ∫y≠0(f (x + y) − f (x) − ∇f (x) ⋅ y1(0,1)(∣y∣))ν(dy) where the L´ evy triplet (b,Q,ν) consists of b ∈ Rd (drift vector), Q ∈ Rd×d symmetric and positive semidefinite (diffusion matrix) a measure ν with ∫y≠0 min{1,∣y∣2}ν(dy) < ∞ (L´ evy measure) Equivalent characterization via the characteristic exponent ψ(ξ) = −ib ⋅ ξ + 1 2ξ ⋅ Qξ + ∫y≠0(1 − eiy⋅ξ + iy ⋅ ξ1(0,1)(∣y∣))ν(dy)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

H¨

• lder–Zygmund space

Cα

b(Rd) ∶=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f ∈ Cb(Rd);∥f ∥Cα

b (Rd) ∶= ∥f ∥∞ + sup

x∈Rd sup 0<∣h∣≤1

∣∆k

hf (x)∣

∣h∣α < ∞ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ where k = ⌊α⌋ + 1 and ∆h ∶= f (x + h) − f (x) ∆k

hf (x) ∶= ∆h(∆k−1 h

f )(x).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

H¨

• lder–Zygmund space

Cα

b(Rd) ∶=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f ∈ Cb(Rd);∥f ∥Cα

b (Rd) ∶= ∥f ∥∞ + sup

x∈Rd sup 0<∣h∣≤1

∣∆k

hf (x)∣

∣h∣α < ∞ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ where k = ⌊α⌋ + 1 and ∆h ∶= f (x + h) − f (x) ∆k

hf (x) ∶= ∆h(∆k−1 h

f )(x). Theorem

1 C k b (Rd) ⊂ Ck b(Rd) for all k ∈ N, 2 C α b (Rd) = Cα b(Rd) for α ∈ (0,∞)/N.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

How to obtain Schauder estimates for solutions to Af = g where A is the generator of a L´ evy process?

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

How to obtain Schauder estimates for solutions to Af = g where A is the generator of a L´ evy process? Known results: Bass ’09, Ros-Oton & Serra ’16: stable operators Bae & Kassmann ’15: ν(dy) = 1/(∣y∣dϕ(y)) classical theory for pseudo-differential operators: ∫∣y∣>1 ∣y∣n ν(dy) < ∞ for n ≫ 1

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

∥f ∥Cα

b (Rd) ≤ c(∥f ∥∞ + ∥Af ∥∞)

Important question: How to measure regularizing effect, i.e. how to find α > 0?

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

∥f ∥Cα

b (Rd) ≤ c(∥f ∥∞ + ∥Af ∥∞)

Important question: How to measure regularizing effect, i.e. how to find α > 0? Idea: Use gradient estimates for the transition density ∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

∥f ∥Cα

b (Rd) ≤ c(∥f ∥∞ + ∥Af ∥∞)

Important question: How to measure regularizing effect, i.e. how to find α > 0? Idea: Use gradient estimates for the transition density ∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1) Equivalent to ∥∇Ptu∥∞ ≤ c′t−1/α∥u∥∞, u ∈ Bb(Rd),t ∈ (0,1).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

Theorem (K. ’19) Let (Xt)t≥0 be a L´ evy process with generator (A,D(A)). Assume that lim

∣ξ∣→∞

Reψ(ξ) log(1 + ∣ξ∣) = ∞ and that the transition density pt satisfies the gradient estimate ∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1) for some α > 0 and c > 0.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

Theorem (K. ’19) Let (Xt)t≥0 be a L´ evy process with generator (A,D(A)). Assume that lim

∣ξ∣→∞

Reψ(ξ) log(1 + ∣ξ∣) = ∞ and that the transition density pt satisfies the gradient estimate ∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1) for some α > 0 and c > 0. Then ∥f ∥Cα

b (Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞) for any f ∈ D(A), Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy generators

Theorem (K. ’19) Let (Xt)t≥0 be a L´ evy process with generator (A,D(A)). Assume that lim

∣ξ∣→∞

Reψ(ξ) log(1 + ∣ξ∣) = ∞ and that the transition density pt satisfies the gradient estimate ∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1) for some α > 0 and c > 0. Then ∥f ∥Cα

b (Rd) ≤ M(∥f ∥∞ + ∥Af ∥∞) for any f ∈ D(A),

∥f ∥Cκ+α

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd)) if Af = g ∈ Cκ

b(Rd) for κ > 0

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/α,

t ∈ (0,1),γ ∈ Nd

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/αemt,

t > 0,γ ∈ Nd

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/αemt,

t > 0,γ ∈ Nd

2 regularizing property of the resolvent Rλu = ∫(0,∞) e−λtPtu dt:

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/αemt,

t > 0,γ ∈ Nd

2 regularizing property of the resolvent Rλu = ∫(0,∞) e−λtPtu dt:

u ∈ Cκ

b(Rd)

⇒ Rλu ∈ Cκ+α

b

(Rd) for all κ ≥ 0 and λ > m

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/αemt,

t > 0,γ ∈ Nd

2 regularizing property of the resolvent Rλu = ∫(0,∞) e−λtPtu dt:

u ∈ Cκ

b(Rd)

⇒ Rλu ∈ Cκ+α

b

(Rd) for all κ ≥ 0 and λ > m , and ∥Rλu∥Cα+κ

b

(Rd) ≤ C∥u∥Cκ

b (Rd),

u ∈ Cκ

b(Rd),κ ≥ 0

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Idea of proof

∫Rd ∣∇pt(x)∣dx ≤ ct−1/α, t ∈ (0,1)

1 Gradient estimate implies

∫Rd ∣∂γ

x pt(x)∣dx ≤ ct−∣γ∣/αemt,

t > 0,γ ∈ Nd

2 regularizing property of the resolvent Rλu = ∫(0,∞) e−λtPtu dt:

u ∈ Cκ

b(Rd)

⇒ Rλu ∈ Cκ+α

b

(Rd) for all κ ≥ 0 and λ > m , and ∥Rλu∥Cα+κ

b

(Rd) ≤ C∥u∥Cκ

b (Rd),

u ∈ Cκ

b(Rd),κ ≥ 0 3 If f ∈ D(A) then f = Rλu for u ∶= λf − Af .

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example I

Example Let (Xt)t≥0 be a L´ evy process with triplet (b,Q,ν) and generator (A,D(A)). If Q is positive definite, then ∥f ∥Cκ+2

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd))

whenever Af = g ∈ Cκ

b(Rd) for some κ ≥ 0. In particular, D(A) ⊆ C2 b(Rd).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Example Let (Xt)t≥0 be a pure-jump L´ evy process with ν(A) ≥ ∫

r0 0 ∫Sd−1 1A(rθ)r−1−α µ(dθ)dr

+ ∫

∞ r0 ∫Sd−1 1A(rθ)r−1−β µ(dθ)dr,

A ∈ B(Rd/{0}) for α ∈ (0,2), β ∈ (0,∞] and a finite measure µ on the unit sphere Sd−1 ⊆ Rd which is non-degenerate.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Example Let (Xt)t≥0 be a pure-jump L´ evy process with ν(A) ≥ ∫

r0 0 ∫Sd−1 1A(rθ)r−1−α µ(dθ)dr

+ ∫

∞ r0 ∫Sd−1 1A(rθ)r−1−β µ(dθ)dr,

A ∈ B(Rd/{0}) for α ∈ (0,2), β ∈ (0,∞] and a finite measure µ on the unit sphere Sd−1 ⊆ Rd which is non-degenerate.Then ∥f ∥Cκ+α

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd))

whenever Af = g ∈ Cκ

b(Rd) for some κ ≥ 0. In particular, D(A) ⊆ Cα b(Rd).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example II

Example Let (Xt)t≥0 be a pure-jump L´ evy process with ν(A) ≥ ∫

r0 0 ∫Sd−1 1A(rθ)r−1−α µ(dθ)dr

+ ∫

∞ r0 ∫Sd−1 1A(rθ)r−1−β µ(dθ)dr,

A ∈ B(Rd/{0}) for α ∈ (0,2), β ∈ (0,∞] and a finite measure µ on the unit sphere Sd−1 ⊆ Rd which is non-degenerate.Then ∥f ∥Cκ+α

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd))

whenever Af = g ∈ Cκ

b(Rd) for some κ ≥ 0. In particular, D(A) ⊆ Cα b(Rd).

Key: gradient estimates by Schilling, Sztonyk & Wang (’12)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example III

Corollary Let (Xt)t≥0 be a pure-jump L´ evy process whose characteristic exponent ψ satisfies the sector condition, ∣Imψ(ξ)∣ ≤ c Reψ(ξ) Reψ(ξ) ≍ ∣ξ∣α for ∣ξ∣ ≫ 1 Then:

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example III

Corollary Let (Xt)t≥0 be a pure-jump L´ evy process whose characteristic exponent ψ satisfies the sector condition, ∣Imψ(ξ)∣ ≤ c Reψ(ξ) Reψ(ξ) ≍ ∣ξ∣α for ∣ξ∣ ≫ 1 Then:

1 Cα+ ∞ (Rd) ⊆ D(A) ⊆ Cα ∞(Rd) where Cα+ ∞ (Rd) ∶= ⋂β>α Cβ ∞(Rd)

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example III

Corollary Let (Xt)t≥0 be a pure-jump L´ evy process whose characteristic exponent ψ satisfies the sector condition, ∣Imψ(ξ)∣ ≤ c Reψ(ξ) Reψ(ξ) ≍ ∣ξ∣α for ∣ξ∣ ≫ 1 Then:

1 Cα+ ∞ (Rd) ⊆ D(A) ⊆ Cα ∞(Rd) where Cα+ ∞ (Rd) ∶= ⋂β>α Cβ ∞(Rd) 2 ∥f ∥Cκ+α

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd)) if Af = g ∈ Cκ

b(Rd) for κ ≥ 0

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Example III

Corollary Let (Xt)t≥0 be a pure-jump L´ evy process whose characteristic exponent ψ satisfies the sector condition, ∣Imψ(ξ)∣ ≤ c Reψ(ξ) Reψ(ξ) ≍ ∣ξ∣α for ∣ξ∣ ≫ 1 Then:

1 Cα+ ∞ (Rd) ⊆ D(A) ⊆ Cα ∞(Rd) where Cα+ ∞ (Rd) ∶= ⋂β>α Cβ ∞(Rd) 2 ∥f ∥Cκ+α

b

(Rd) ≤ M(∥f ∥∞ + ∥Af ∥Cκ

b (Rd)) if Af = g ∈ Cκ

b(Rd) for κ ≥ 0 3 D(A) is an algebra, i.e. f ,g ∈ D(A) implies f ⋅ g ∈ D(A), and

A(f ⋅ g) = fAg + gAf + Γ(f ,g) with Γ(f ,g)(x) ∶= ∫y≠0(f (x + y) − f (x))(g(x + y) − g(x))ν(dy).

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Remarks

The presented Schauder estimates . . . are sharp. . . . hold more generally for functions f in the Favard space of order 1, F1 ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f ∈ Bb(Rd); sup

t∈(0,1)

sup

x∈Rd

∣Ef (x + Xt) − f (x)∣ t < ∞ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for L´ evy-type operators

Q: Can the results be extended to L´ evy-type operators Af (x) = b(x) ⋅ ∇f (x) + 1 2 tr(Q(x) ⋅ ∇2f (x)) + ∫y≠0(f (x + y) − f (x) − ∇f (x) ⋅ y1(0,1)(∣y∣))ν(x,dy) . . . ?

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for operators of variable order

Theorem (K. ’19) Consider Af (x) = c(x)∫y≠0(f (x + y) − f (x) − y∇f (x)1(0,1)(∣y∣)) 1 ∣y∣d+α(x) dy for α ∶ Rd → (0,2) H¨

• lder continuous with infx α(x) > 0.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for operators of variable order

Theorem (K. ’19) Consider Af (x) = c(x)∫y≠0(f (x + y) − f (x) − y∇f (x)1(0,1)(∣y∣)) 1 ∣y∣d+α(x) dy for α ∶ Rd → (0,2) H¨

• lder continuous with infx α(x) > 0. Then the

Schauder estimate ∥f ∥Cα(⋅)−ε

b

(Rd) ≤ Mε(∥f ∥∞ + ∥Af ∥∞),

f ∈ D(A) holds for ε > 0; here Cα(⋅)−ε

b

(Rd) is a H¨

• lder–Zygmund space of variable
• rder.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

Schauder estimates for operators of variable order

Theorem (K. ’19) Consider Af (x) = c(x)∫y≠0(f (x + y) − f (x) − y∇f (x)1(0,1)(∣y∣)) 1 ∣y∣d+α(x) dy for α ∶ Rd → (0,2) H¨

• lder continuous with infx α(x) > 0. Then the

Schauder estimate ∥f ∥Cα(⋅)−ε

b

(Rd) ≤ Mε(∥f ∥∞ + ∥Af ∥∞),

f ∈ D(A) holds for ε > 0; here Cα(⋅)−ε

b

(Rd) is a H¨

• lder–Zygmund space of variable
• rder. In particular,

D(A) ⊆ Cα(⋅)−ε

b

(Rd), ε > 0.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

References I

Bae, J., Kassmann, M.: Schauder estimates in generalized H¨

• lder
• spaces. arXiv 1505.05498.

Bass, R. F.: Regularity results for stable-like operators. J. Funct. Anal. 257 (2009), 2693–2722. Knopova, V., Schilling, R.L.: A note on the existence of transition probability densities of L´ evy processes. Forum. Math. 25 (2013), 125–149. K¨ uhn, F.: Schauder Estimates for Equations Associated with L´ evy

• Generators. Int. Eq. Op. Theory 91:10 (2019).

K¨ uhn, F.: Schauder estimates for Poisson equations associated with non-local Feller generators. arXiv 1902.01760.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators

Introduction Definitions Results Outlook

References II

K¨ uhn, F., Schilling, R.L.: On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal. 276 (2019), 2397–2439. Ros-Oton, X., Serra, J.: Regularity theory for general stable operators.

• J. Diff. Equations 260 (2016), 8675–8715.

Schilling, R. L., Sztonyk, P., Wang, J.: Coupling property and gradient estimates of L´ evy processes via the symbol. Bernoulli 18 (2012), 1128–1149. Stein, E. M.: Singular integrals and differentiability properties of

• functions. Princeton Univ. Press 1970.

Triebel, H.: Interpolation theory, function spaces, differential

• perators. North-Holland 1978.

Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators