(Centered) Hardy–Littlewood maximal function

Regularity of the local fractional maximal function

Janne Korvenp¨a¨a

June 12 2015

Workshop on harmonic analysis and partial differential equations Aalto University

Based on a joint work with T. Heikkinen, J. Kinnunen and H. Tuominen.

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(Centered) Hardy–Littlewood maximal function

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(Centered) Hardy–Littlewood maximal function

Mu(x) = sup

r>0

Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ.

2 / 16

(Centered) Hardy–Littlewood maximal function

Mu(x) = sup

r>0

Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ.

By Hardy–Littlewood–Wiener theorem, kMuk_L^p₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1.

2 / 16

(Centered) Hardy–Littlewood maximal function

Mu(x) = sup

r>0

Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ.

By Hardy–Littlewood–Wiener theorem, kMuk_L^p₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1.

If u ∈ W ^1,p(Rⁿ), then |DMu(x)| ≤ M |Du|(x) for a.e. x ∈ Rⁿ [J. Kinnunen, 1997].

2 / 16

(Centered) Hardy–Littlewood maximal function

Mu(x) = sup

r>0

Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ.

By Hardy–Littlewood–Wiener theorem, kMuk_L^p₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1.

If u ∈ W ^1,p(Rⁿ), then |DMu(x)| ≤ M |Du|(x) for a.e. x ∈ Rⁿ [J. Kinnunen, 1997].

Consequently, M: W ^1,p(Rⁿ) → W ^1,p(Rⁿ).

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Local Hardy–Littlewood maximal function

3 / 16

Local Hardy–Littlewood maximal function

M_Ωu(x) = sup Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

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Local Hardy–Littlewood maximal function

M_Ωu(x) = sup Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case, kM_Ωuk_L^p_(Ω) ≤ Ckuk_L^p_(Ω), p > 1.

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Local Hardy–Littlewood maximal function

M_Ωu(x) = sup Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case, kM_Ωuk_L^p_(Ω) ≤ Ckuk_L^p_(Ω), p > 1.

If u ∈ W ^1,p(Ω), then |DM_Ωu(x)| ≤ 2M_Ω|Du|(x) for a.e. x ∈ Ω [J. Kinnunen, P. Lindqvist, 1998].

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Local Hardy–Littlewood maximal function

M_Ωu(x) = sup Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case, kM_Ωuk_L^p_(Ω) ≤ Ckuk_L^p_(Ω), p > 1.

If u ∈ W ^1,p(Ω), then |DM_Ωu(x)| ≤ 2M_Ω|Du|(x) for a.e. x ∈ Ω [J. Kinnunen, P. Lindqvist, 1998].

Consequently, M_Ω : W ^1,p(Ω) → W ^1,p(Ω).

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Fractional maximal function; Sobolev spaces

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Fractional maximal function; Sobolev spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

4 / 16

Fractional maximal function; Sobolev spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Boundedness from L^p to L^p∗: kM_αuk_L^p∗₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

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Fractional maximal function; Sobolev spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Boundedness from L^p to L^p∗: kM_αuk_L^p∗₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

If u ∈ W ^1,p(Rⁿ) and 0 ≤ α < n/p, then |DM_αu(x)| ≤ M_α|Du|(x) for a.e. x ∈ Rⁿ [J. Kinnunen, E. Saksman, 2003].

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Fractional maximal function; Sobolev spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Boundedness from L^p to L^p∗: kM_αuk_L^p∗₍Rⁿ) ≤ Ckuk_L^p₍Rⁿ), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

If u ∈ W ^1,p(Rⁿ) and 0 ≤ α < n/p, then |DM_αu(x)| ≤ M_α|Du|(x) for a.e. x ∈ Rⁿ [J. Kinnunen, E. Saksman, 2003].

Consequently, M_α : W^1,p(Rⁿ) → W ^1,p∗(Rⁿ).

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Fractional maximal function; Lebesgue spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

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Fractional maximal function; Lebesgue spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Even if we only assume u ∈ L^p(Rⁿ), we get an estimate for |DM_αu|.

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Fractional maximal function; Lebesgue spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Even if we only assume u ∈ L^p(Rⁿ), we get an estimate for |DM_αu|.

If u ∈ L^p(Rⁿ) and 1 ≤ α < n/p, then |DM_αu(x)| ≤ CM_α−1 u(x) for a.e. x ∈ Rⁿ [J. Kinnunen, E. Saksman, 2003].

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Fractional maximal function; Lebesgue spaces

M_αu(x) = sup

r>0

r^α Z

B(x,r)

|u(y)| dy, x ∈ Rⁿ, α ≥ 0.

Even if we only assume u ∈ L^p(Rⁿ), we get an estimate for |DM_αu|.

If u ∈ L^p(Rⁿ) and 1 ≤ α < n/p, then |DM_αu(x)| ≤ CM_α−1 u(x) for a.e. x ∈ Rⁿ [J. Kinnunen, E. Saksman, 2003].

Consequently, |DM_αu| ∈ L^q(Rⁿ), q = np/(n − (α − 1)p).

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Local fractional maximal function

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Local fractional maximal function

M_α,Ωu(x) = sup r^α Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, α ≥ 0, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

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Local fractional maximal function

M_α,Ωu(x) = sup r^α Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, α ≥ 0, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case: kM_α,Ωuk_L^p∗_(Ω) ≤ Ckuk_L^p_(Ω), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

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Local fractional maximal function

M_α,Ωu(x) = sup r^α Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, α ≥ 0, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case: kM_α,Ωuk_L^p∗_(Ω) ≤ Ckuk_L^p_(Ω), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

One might expect that similar estimates hold for the weak gradient as in the global case.

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Local fractional maximal function

M_α,Ωu(x) = sup r^α Z

B(x,r)

|u(y)| dy, x ∈ Ω ⊂ Rⁿ, α ≥ 0, where the supremum is taken over {0 < r < dist(x, Rⁿ \ Ω)}.

As in the global case: kM_α,Ωuk_L^p∗_(Ω) ≤ Ckuk_L^p_(Ω), p > 1, 0 ≤ α < n/p, p^∗ = np/(n − αp).

One might expect that similar estimates hold for the weak gradient as in the global case.

However, there will be extra terms occurring in our estimates.

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Spherical (fractional) maximal function

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Spherical (fractional) maximal function

S_αu(x) = sup

r>0

r^α Z

∂B(x,r)

|u(y)| dHⁿ⁻¹(y), x ∈ Rⁿ, α ≥ 0,

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Spherical (fractional) maximal function

S_αu(x) = sup

r>0

r^α Z

∂B(x,r)

|u(y)| dHⁿ⁻¹(y), x ∈ Rⁿ, α ≥ 0, Local version appears in our estimates for |DM_α,Ωu|.

7 / 16

Spherical (fractional) maximal function

S_αu(x) = sup

r>0

r^α Z

∂B(x,r)

|u(y)| dHⁿ⁻¹(y), x ∈ Rⁿ, α ≥ 0, Local version appears in our estimates for |DM_α,Ωu|.

Boundedness from L^p to L^p∗: if p > n/(n − 1) and 0 ≤ α < minn

n−1

p , n − _(n−²ⁿ_1)po

, then kS_αuk_L^p∗_(Rⁿ₎ ≤ Ckuk_L^p₍Rⁿ)

[E.M. Stein, 1976] (α = 0, n ≥ 3), [J. Bourgain, 1986] (α = 0, n = 2),

[W. Schlag, C. Sogge, 1997] (α > 0, n ≥ 3), [W. Schlag, 1997] (α > 0, n = 2).

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Spherical (fractional) maximal function

S_αu(x) = sup

r>0

r^α Z

∂B(x,r)

|u(y)| dHⁿ⁻¹(y), x ∈ Rⁿ, α ≥ 0, Local version appears in our estimates for |DM_α,Ωu|.

Boundedness from L^p to L^p∗: if p > n/(n − 1) and 0 ≤ α < minn

n−1

p , n − _(n−²ⁿ_1)po

, then kS_αuk_L^p∗_(Rⁿ₎ ≤ Ckuk_L^p₍Rⁿ)

[E.M. Stein, 1976] (α = 0, n ≥ 3), [J. Bourgain, 1986] (α = 0, n = 2),

[W. Schlag, C. Sogge, 1997] (α > 0, n ≥ 3), [W. Schlag, 1997] (α > 0, n = 2).

The same estimate also holds for the local version S_α,Ωu.

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Local fractional maximal function; Lebesgue spaces

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Local fractional maximal function; Lebesgue spaces

Theorem 1 (Heikkinen, Kinnunen, K, Tuominen)

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < minn

n−1

p , n − _(n−²ⁿ_1)po

+ 1. If u ∈ L^p(Ω), then |DM_α,Ωu| ∈ L^q(Ω) with q = np/(n − (α − 1)p).

Moreover,

|DM_α,Ωu(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

for almost every x ∈ Ω, where the constant C depends only on n.

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Local fractional maximal function; Lebesgue spaces

Theorem 1 (Heikkinen, Kinnunen, K, Tuominen)

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < minn

n−1

p , n − _(n−²ⁿ_1)po

+ 1. If u ∈ L^p(Ω), then |DM_α,Ωu| ∈ L^q(Ω) with q = np/(n − (α − 1)p).

Moreover,

|DM_α,Ωu(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

for almost every x ∈ Ω, where the constant C depends only on n.

Additional term containing local spherical fractional function compared to the global case.

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Main idea of the proof

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Main idea of the proof

1. When u ∈ L^p(Ω) ∩ C^∞(Ω), estimate

|Du_t^α(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

, 0 < t < 1, (1) for fractional average functions

u_t^α(x) = (tδ(x))^α Z

B(x,tδ(x))

u(y)dy, δ(x) = dist(x, Rⁿ \ Ω).

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Main idea of the proof

1. When u ∈ L^p(Ω) ∩ C^∞(Ω), estimate

|Du_t^α(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

, 0 < t < 1, (1) for fractional average functions

u_t^α(x) = (tδ(x))^α Z

B(x,tδ(x))

u(y)dy, δ(x) = dist(x, Rⁿ \ Ω).

2. When u ∈ L^p(Ω), estimate (1) by approximation.

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Main idea of the proof

1. When u ∈ L^p(Ω) ∩ C^∞(Ω), estimate

|Du_t^α(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

, 0 < t < 1, (1) for fractional average functions

u_t^α(x) = (tδ(x))^α Z

B(x,tδ(x))

u(y)dy, δ(x) = dist(x, Rⁿ \ Ω).

2. When u ∈ L^p(Ω), estimate (1) by approximation.

3. Local fractional maximal function can be approximated pointwise by v_k = max_1≤j≤k |u|^α_tj, where t_j is an enumeration of the rationals

between 0 and 1.

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Main idea of the proof

1. When u ∈ L^p(Ω) ∩ C^∞(Ω), estimate

|Du_t^α(x)| ≤ C M_α−1,Ωu(x) + S_α−1,Ωu(x)

, 0 < t < 1, (1) for fractional average functions

u_t^α(x) = (tδ(x))^α Z

B(x,tδ(x))

u(y)dy, δ(x) = dist(x, Rⁿ \ Ω).

2. When u ∈ L^p(Ω), estimate (1) by approximation.

3. Local fractional maximal function can be approximated pointwise by v_k = max_1≤j≤k |u|^α_tj, where t_j is an enumeration of the rationals

between 0 and 1.

4. We can extract a subsequence {v_kj } such that |Dv_kj| converges weakly to |DM_α,Ωu| in L^q(Ω) as j → ∞.

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Mapping properties; Lebesgue spaces

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Mapping properties; Lebesgue spaces

If Ω has finite measure, then M_α,Ω : L^p(Ω) → W ^1,q(Ω).

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Mapping properties; Lebesgue spaces

If Ω has finite measure, then M_α,Ω : L^p(Ω) → W ^1,q(Ω).

Corollary 2

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ L^p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p).

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Mapping properties; Lebesgue spaces

If Ω has finite measure, then M_α,Ω : L^p(Ω) → W ^1,q(Ω).

Corollary 2

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ L^p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p).

Moreover, M_α,Ωu has zero boundary values in Sobolev’s sense since Z

Ω

M_α,Ωu(x) dist(x, Rⁿ \ Ω)

q

dx < ∞.

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Mapping properties; Lebesgue spaces

If Ω has finite measure, then M_α,Ω : L^p(Ω) → W ^1,q(Ω).

Corollary 2

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ L^p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p).

Moreover, M_α,Ωu has zero boundary values in Sobolev’s sense since Z

Ω

M_α,Ωu(x) dist(x, Rⁿ \ Ω)

q

dx < ∞.

Corollary 3

Let n ≥ 2, p > n/(n − 1) and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ L^p(Ω), then M_α,Ωu ∈ W₀^1,q(Ω) with q = np/(n − (α − 1)p).

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Local fractional maximal function; Sobolev spaces

11 / 16

Local fractional maximal function; Sobolev spaces

Local fractional maximal function of a Sobolev function is not necessarily smoother than of an arbitrary function in L^p(Ω).

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Local fractional maximal function; Sobolev spaces

Local fractional maximal function of a Sobolev function is not necessarily smoother than of an arbitrary function in L^p(Ω).

Theorem 4 (Heikkinen, Kinnunen, K, Tuominen)

Let n ≥ 2, 1 < p < n and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ W ^1,p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p). Moreover,

|DM_α,Ωu(x)| ≤ 2M_α,Ω|Du|(x) + αM_α−1,Ωu(x) for almost every x ∈ Ω.

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Local fractional maximal function; Sobolev spaces

Local fractional maximal function of a Sobolev function is not necessarily smoother than of an arbitrary function in L^p(Ω).

Theorem 4 (Heikkinen, Kinnunen, K, Tuominen)

Let n ≥ 2, 1 < p < n and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ W ^1,p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p). Moreover,

|DM_α,Ωu(x)| ≤ 2M_α,Ω|Du|(x) + αM_α−1,Ωu(x) for almost every x ∈ Ω.

However, for regular domains we obtain the same regularity as in Rⁿ.

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Local fractional maximal function; Sobolev spaces

Local fractional maximal function of a Sobolev function is not necessarily smoother than of an arbitrary function in L^p(Ω).

Theorem 4 (Heikkinen, Kinnunen, K, Tuominen)

Let n ≥ 2, 1 < p < n and let 1 ≤ α < n/p. If |Ω| < ∞ and u ∈ W ^1,p(Ω), then M_α,Ωu ∈ W ^1,q(Ω) with q = np/(n − (α − 1)p). Moreover,

|DM_α,Ωu(x)| ≤ 2M_α,Ω|Du|(x) + αM_α−1,Ωu(x) for almost every x ∈ Ω.

However, for regular domains we obtain the same regularity as in Rⁿ.

Corollary 5

If Ω is bounded with a C¹-boundary, then Theorem 4 holds with a better exponent p^∗ = np/(n − αp) instead of q.

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Example 1

12 / 16

Example 1

In general, the estimate

|DM_α,Ωu(x)| ≤ CM_α−1,Ωu(x) for a.e. x ∈ Ω (2) cannot hold in the local case; S_α−1,Ωu(x) is needed in Theorem 1.

12 / 16

Example 1

In general, the estimate

|DM_α,Ωu(x)| ≤ CM_α−1,Ωu(x) for a.e. x ∈ Ω (2) cannot hold in the local case; S_α−1,Ωu(x) is needed in Theorem 1.

Let n ≥ 2 and Ω = B(0, 1) ⊂ Rⁿ. Let 1 < p < ∞, α ≥ 1 and let 0 < β < 1. Then the function u,

u(x) = (1 − |x|)^−β/p, belongs to L^p(Ω).

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Example 1

In general, the estimate

|DM_α,Ωu(x)| ≤ CM_α−1,Ωu(x) for a.e. x ∈ Ω (2) cannot hold in the local case; S_α−1,Ωu(x) is needed in Theorem 1.

Let n ≥ 2 and Ω = B(0, 1) ⊂ Rⁿ. Let 1 < p < ∞, α ≥ 1 and let 0 < β < 1. Then the function u,

u(x) = (1 − |x|)^−β/p, belongs to L^p(Ω).

When |x| is small enough, the maximizing radius for M_α,Ωu(x) is the largest possible, i.e. 1 − |x|.

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Example 1

In general, the estimate

|DM_α,Ωu(x)| ≤ CM_α−1,Ωu(x) for a.e. x ∈ Ω (2) cannot hold in the local case; S_α−1,Ωu(x) is needed in Theorem 1.

Let n ≥ 2 and Ω = B(0, 1) ⊂ Rⁿ. Let 1 < p < ∞, α ≥ 1 and let 0 < β < 1. Then the function u,

u(x) = (1 − |x|)^−β/p, belongs to L^p(Ω).

When |x| is small enough, the maximizing radius for M_α,Ωu(x) is the largest possible, i.e. 1 − |x|.

For those x, |DM_α,Ωu(x)| consists of an average term over a ball and one over a sphere, where the former is uniformly bounded and the

latter tends to ∞ as |x| → 0.

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Example 2

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Example 2

There are domains Ω ⊂ Rⁿ, for which M_α,Ω(W ^1,p(Ω)) 6⊂ W ^1,ˆq(Ω) when ˆq > q = np/(n − (α − 1)p); The exponent q in Theorem 4 is sharp.

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Example 2

There are domains Ω ⊂ Rⁿ, for which M_α,Ω(W ^1,p(Ω)) 6⊂ W ^1,ˆq(Ω) when ˆq > q = np/(n − (α − 1)p); The exponent q in Theorem 4 is sharp.

Let n ≥ 2, α ≥ 1 and (α − 1)p < n. Let Ω = int [^∞

k=1

Q_k ∪ C_k ,

Q_k = [k, k + 2^−k]×[0, 2^−k]ⁿ⁻¹, C_k = [k + 2^−k, k + 1]×[0, 2^−3k]ⁿ⁻¹.

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Example 2

There are domains Ω ⊂ Rⁿ, for which M_α,Ω(W ^1,p(Ω)) 6⊂ W ^1,ˆq(Ω) when ˆq > q = np/(n − (α − 1)p); The exponent q in Theorem 4 is sharp.

Let n ≥ 2, α ≥ 1 and (α − 1)p < n. Let Ω = int [^∞

k=1

Q_k ∪ C_k ,

Q_k = [k, k + 2^−k]×[0, 2^−k]ⁿ⁻¹, C_k = [k + 2^−k, k + 1]×[0, 2^−3k]ⁿ⁻¹. For every ˆq > q = np/(n − (α − 1)p) we define a function u such

that u = 2^kn/ˆp on Q_k and u increases linearly from 2^kn/ˆp to 2^(k+1)n/ˆp on C_k, where ˆp = nˆq/(n + (α − 1)ˆq) > p.

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Example 2

There are domains Ω ⊂ Rⁿ, for which M_α,Ω(W ^1,p(Ω)) 6⊂ W ^1,ˆq(Ω) when ˆq > q = np/(n − (α − 1)p); The exponent q in Theorem 4 is sharp.

Let n ≥ 2, α ≥ 1 and (α − 1)p < n. Let Ω = int [^∞

k=1

Q_k ∪ C_k ,

Q_k = [k, k + 2^−k]×[0, 2^−k]ⁿ⁻¹, C_k = [k + 2^−k, k + 1]×[0, 2^−3k]ⁿ⁻¹. For every ˆq > q = np/(n − (α − 1)p) we define a function u such

that u = 2^kn/ˆp on Q_k and u increases linearly from 2^kn/ˆp to 2^(k+1)n/ˆp on C_k, where ˆp = nˆq/(n + (α − 1)ˆq) > p.

Then u ∈ W ^1,p(Ω) but

|DM_α,Ωu| 6∈ L^qˆ(Ω).

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Example 3

14 / 16

Example 3

In the case 0 < α < 1, M_α,Ωu can be very irregular, even when u is a constant function; The lower bound α ≥ 1 is sharp in Theorems 1 and 4.

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Example 3

In the case 0 < α < 1, M_α,Ωu can be very irregular, even when u is a constant function; The lower bound α ≥ 1 is sharp in Theorems 1 and 4.

Let n ≥ 1, 0 < α < 1 and r > 0. Let β be an integer satisfying β ≥ n/((1 − α)r), and let

Ω = B(0, 2) \ [

k≥1

S_k, S_k = {2^−k + j2^−(1+β)k : j = 1, . . . , 2^βk}ⁿ.

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Example 3

In the case 0 < α < 1, M_α,Ωu can be very irregular, even when u is a constant function; The lower bound α ≥ 1 is sharp in Theorems 1 and 4.

Let n ≥ 1, 0 < α < 1 and r > 0. Let β be an integer satisfying β ≥ n/((1 − α)r), and let

Ω = B(0, 2) \ [

k≥1

S_k, S_k = {2^−k + j2^−(1+β)k : j = 1, . . . , 2^βk}ⁿ. Then |DM_α,Ωu| for u ≡ 1 does not belong to L^r(Ω).

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References

T. Heikkinen, J. Kinnunen, J. Korvenp¨a¨a and H. Tuominen, Regularity of the local fractional maximal function, Ark. Mat.

53 (2015), 127-154.

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986), 69–85.

J. Kinnunen, The Hardy–Littlewood maximal function of a Sobolev-function, Israel J.Math. 100 (1997), 117–124.

J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161–167.

J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529–535.

W. Schlag, A generalization of Bourgain’s circular maximal theorem J. Amer. Math. Soc. 10, no. 1 (1997), 103–122.

W. Schlag and C. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997), no.1, 1–15.

E.M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad.

Sci. U.S.A. 73 (1976), no. 7, 2174–2175.

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THANK YOU

FOR YOUR ATTENTION

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