Weighted norm inequalities for integral transforms with kernels - - PowerPoint PPT Presentation

Weighted norm inequalities for integral transforms with kernels bounded by power functions

Alberto Debernardi

Centre de Recerca Matem` atica, Barcelona 6th Workshop on Fourier Analysis and Related Topics University of P´ ecs, 26 August 2017

• A. Debernardi–CRM

WNI’s for integral transforms 1 / 22

Brief history: weighted norm inequalities for the Fourier transform

For the Fourier transform

• f(y) =
• R

f(x)e−ixy dx, it was proved in the 1980s (Muckenhoupt, Jurkat-Sampson) that the weighted norm inequality fq,u :=

R

u(y)| f(y)|q dy 1/q ≤ C

R

v(x)|f(x)|p dx 1/p =: fp,v holds for every f with 1 < p ≤ q < ∞ and C > 0 independent of f provided that there exists D > 0 such that for every r > 0 1/r u∗(y) dy 1/q r (1/v)∗(x)1−p′ dx 1/p′ ≤ D.

• A. Debernardi–CRM

WNI’s for integral transforms 2 / 22

Problem

Given an integral transform Ff(y) = yc0 ∞ xb0f(x)K(x, y) dx, y > 0, b0, c0 ∈ R, where |K(x, y)| min

• (xy)b1, (xy)b2

, b1 > b2. We also assume 1

0 xb0+b1|f(x)| dx +

∞

1 xb0+b2|f(x)| dx < ∞.

We want to give sufficient (and necessary, when possible) conditions for the weighted norm inequality y−βFfq ≤ Cxγfp, 1 < p ≤ q < ∞, (1) to hold, using an approach that does not involve decreasing rearrangements.

• A. Debernardi–CRM

WNI’s for integral transforms 3 / 22

Examples

• 1. The Fourier transform is not a good example, since

|K(x, y)| = |e2πixy| = 1 does not satisfy |K(x, y)| min

• (xy)b1, (xy)b2

, b1 > b2. The cosine transform is also a bad example.

• 2. The sine transform satisfies

|K(x, y)| = | sin xy| min

• xy, 1
• ,

i.e., b1 = 1 > 0 = b2.

• A. Debernardi–CRM

WNI’s for integral transforms 4 / 22

Examples: The Hankel transform

• 3. The classical Hankel transform of order α ≥ −1/2 is defined as

Hαf(y) = ∞ x2α+1f(x)jα(xy) dx, where jα is the normalized Bessel function of order α, represented through the series jα(z) = Γ(α + 1)

∞

• n=0

(−1)n(z/2)2n n!Γ(n + α + 1). The function jα satisfies the estimate |jα(xy)| min

• 1, (xy)−α−1/2}.
• A. Debernardi–CRM

WNI’s for integral transforms 5 / 22

Examples: The Hα transform

• 4. The so-called Hα transform is defined as

Hαf(y) = ∞ (xy)1/2f(x)Hα(xy) dx, α > −1/2, where Hα is the Struve function, defined as Hα(z) = z 2 α+1 ∞

• n=0

(−1)n(z/2)2n Γ(n + 3/2)Γ(n + α + 3/2).

• A. Debernardi–CRM

WNI’s for integral transforms 6 / 22

Examples: The Hα transform

• 4. The so-called Hα transform is defined as

Hαf(y) = ∞ (xy)1/2f(x)Hα(xy) dx, α > −1/2, where Hα is the Struve function, defined as Hα(z) = z 2 α+1 ∞

• n=0

(−1)n(z/2)2n Γ(n + 3/2)Γ(n + α + 3/2). The Struve function satisfies the estimate |Hα(x)|

• min{xα+1, x−1/2},

α < 1/2, min{xα+1, xα−1}, α ≥ 1/2,

◮ P. G. Rooney, Canad. J. Math (1980).

• A. Debernardi–CRM

WNI’s for integral transforms 6 / 22

Known results

Cosine transform (and Fourier transform): if Ff = f or Ff = fcos, then (1) holds if and only if β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 1/q.

◮ W. B. Jurkat–G. Sampson, Indiana Univ. Math. J. (1984); B.

Muckenhoupt, Proc. Amer. Math. Soc. (1983).

◮ H. P. Heinig, Indiana Univ. Math. J. (1984).

• A. Debernardi–CRM

WNI’s for integral transforms 7 / 22

Known results

Cosine transform (and Fourier transform): if Ff = f or Ff = fcos, then (1) holds if and only if β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 1/q.

◮ W. B. Jurkat–G. Sampson, Indiana Univ. Math. J. (1984); B.

Muckenhoupt, Proc. Amer. Math. Soc. (1983).

◮ H. P. Heinig, Indiana Univ. Math. J. (1984).

Sine transform: if Ff = fsin, (1) holds if and only if β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 1 + 1/q.

◮ D. Gorbachev, E. Liflyand, S. Tikhonov, Indiana Univ. Math. J.

(to appear).

• A. Debernardi–CRM

WNI’s for integral transforms 7 / 22

Known results

Hankel transform: if Ff = Hαf (α ≥ −1/2), then (1) holds if and

• nly if β = γ − 2α − 1 + 1/q − 1/p′ and

max{1/q − 1/p′, 0} − α − 1/2 ≤ β < 1/q.

◮ P. Heywood, P. G. Rooney, Proc. Roy. Soc. Edinburgh (1984). ◮ L. de Carli, J. Math. Anal. Appl. (2008).

• A. Debernardi–CRM

WNI’s for integral transforms 8 / 22

Known results

Hα transform: if Ff = Hαf (α > −1/2), (1) holds provided that β = γ + 1/q − 1/p′ and

– for −1/2 < α < 1/2, β ≥ max{1/q − 1/p′, 0} and 1/q + α − 1/2 < β < 1/q + α + 3/2; – for α ≥ 1/2, 1/q + α − 1/2 < β < 1/q + α + 3/2.

◮ P. G. Rooney, Canad. J. Math. (1980).

• A. Debernardi–CRM

WNI’s for integral transforms 9 / 22

Main result (sufficient conditions)

The following states sufficient conditions for inequality (1) to hold.

Theorem

Let 1 < p ≤ q < ∞. If the integral transform Ff(y) = yc0 ∞ xb0f(x)K(x, y) dx, y > 0, b0, c0 ∈ R, satisfies |K(x, y)| min

• (xy)b1, (xy)b2

, with b1 > b2, then the inequality y−βFfq ≤ Cxγfp, 1 < p ≤ q < ∞, holds for every f, provided that β = γ + c0 − b0 + 1 q − 1 p′ , 1 q + c0 + b2 < β < 1 q + c0 + b1.

• A. Debernardi–CRM

WNI’s for integral transforms 10 / 22

Sharpness and necessity conditions

If instead of |K(x, y)| min

• (xy)b1, (xy)b2

, there holds K(x, y) ≍ min

• (xy)b1, (xy)b2

, the latter theorem can be improved.

Theorem

Let 1 < p ≤ q < ∞. If the integral transform Ff(y) = yc0 ∞ xb0f(x)K(x, y) dx, y > 0, b0, c0 ∈ R, satisfies |K(x, y)| min

• (xy)b1, (xy)b2

, with b1 > b2, then the inequality y−βFfq ≤ Cxγfp holds for every f with β = γ + c0 − b0 + 1 q − 1 p′ , 1 q + c0 + b2 < β < 1 q + c0 + b1.

• A. Debernardi–CRM

WNI’s for integral transforms 11 / 22

Sharpness and necessity conditions

If instead of |K(x, y)| min

• (xy)b1, (xy)b2

, there holds K(x, y) ≍ min

• (xy)b1, (xy)b2

, the latter theorem can be improved.

Theorem

Let 1 < p ≤ q < ∞. If the integral transform Ff(y) = yc0 ∞ xb0f(x)K(x, y) dx, y > 0, b0, c0 ∈ R, satisfies K(x, y) ≍ min

• (xy)b1, (xy)b2

, with b1 > b2, then the inequality y−βFfq ≤ Cxγfp holds for every f if and only if β = γ + c0 − b0 + 1 q − 1 p′ , 1 q + c0 + b2 < β < 1 q + c0 + b1.

• A. Debernardi–CRM

WNI’s for integral transforms 11 / 22

Examples

We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! (b1 > b2). max{1/q − 1/p′, 0} ≤ β < 1/q.

• A. Debernardi–CRM

WNI’s for integral transforms 12 / 22

Examples

We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! (b1 > b2). max{1/q − 1/p′, 0} ≤ β < 1/q. Sine transform: 1/q < β < 1 + 1/q max{1/q − 1/p′, 0} ≤ β < 1 + 1/q.

• A. Debernardi–CRM

WNI’s for integral transforms 12 / 22

Examples

We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! (b1 > b2). max{1/q − 1/p′, 0} ≤ β < 1/q. Sine transform: 1/q < β < 1 + 1/q max{1/q − 1/p′, 0} ≤ β < 1 + 1/q. Hankel transform of order α > −1/2: 1/q − α − 1/2 < β < 1/q max{1/q − 1/p′, 0} − α − 1/2 ≤ β < 1/q.

• A. Debernardi–CRM

WNI’s for integral transforms 12 / 22

Examples

Hα transform (α > −1/2): 1/q < β < 1/q + α + 3/2, −1/2 < α < 1/2, 1/q + α − 1/2 < β < 1/q + α + 3/2, α ≥ 1/2. Recall the known sufficient conditions: – For −1/2 < α < 1/2, β ≥ max{1/q − 1/p′, 0} and 1/q + α − 1/2 < β < 1/q + α + 3/2. – For α ≥ 1/2, 1/q + α − 1/2 < β < 1/q + α + 3/2.

• A. Debernardi–CRM

WNI’s for integral transforms 13 / 22

Transforms with kernel represented by power series

Recall that y−β fq ≤ Cxγfp (2) holds for every f if and only if β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 1/q. (3) It was proved by Sadosky and Wheeden that if

• R f = 0, then (2) holds

for 1/q < β < 1 + 1/q, additionally to (3).

◮ C. Sadosky and R. L. Wheeden, Trans. Amer. Mat. Soc. (1987).

• A. Debernardi–CRM

WNI’s for integral transforms 14 / 22

Transforms with kernel represented by power series

Assume that the kernel of the transform F is expressed as K(x, y) = (xy)b1

∞

• m=0

am(xy)km, k ∈ N, am ∈ C. If |K(x, y)| min

• (xy)b1, (xy)b2

, with b1 ≥ b2, we write Ff(y) = yc0 ∞ xb0(xy)b1f(x)

• (xy)−b1K(x, y)
• dx,

It is clear that (xy)−b1|K(x, y)| min

• 1, (xy)b2−b1

.

• A. Debernardi–CRM

WNI’s for integral transforms 15 / 22

Transforms with kernel represented by power series

If f is a function such that ∞ xb0+b1+kℓf(x) dx = 0, ℓ = 0, . . . , n − 1, n ∈ N, (4) then we can write Ff(y) = yc0 ∞ xb0(xy)b1f(x)

• (xy)−b1K(x, y)

−

m−1

• ℓ=0

aℓ(xy)kℓ

• dx,

for any m = 1, . . . , n.

• A. Debernardi–CRM

WNI’s for integral transforms 16 / 22

Transforms with kernel represented by power series

Since (xy)−b1|K(x, y)| min

• 1, (xy)b2−b1

, with b1 ≥ b2, if we define Gℓ(x, y) := (xy)−b1K(x, y) −

ℓ−1

• m=0

am(xy)km, the following estimate holds: |Gℓ(x, y)| min

• (xy)kℓ, (xy)k(ℓ−1)

. In view of the latter estimate, we can apply our main result.

• A. Debernardi–CRM

WNI’s for integral transforms 17 / 22

Transforms with kernel represented by power series

Theorem

Let 1 < p ≤ q < ∞ and assume the kernel K is given by K(x, y) = (xy)b1

∞

• m=0

am(xy)km, k ∈ N, am ∈ C. Furthermore, suppose that |K(x, y)| min

• (xy)b1, (xy)b2

, with b1 ≥ b2. Then, the weighted norm inequality y−βFfq xγfp holds for all f satisfying (4) with β = γ + c0 − b0 + 1/q − 1/p′ and 1/q + c0 + b1 < β < 1/q + c0 + b1 + nℓ, where β = 1/q + c0 + b1 + kℓ for ℓ = 1, . . . , n − 1.

• A. Debernardi–CRM

WNI’s for integral transforms 18 / 22

Application

As a simple example of the latter, consider f with ∞

0 f(x) dx = 0.

Since cos xy =

∞

• n=0

(−1)n(xy)2n (2n)! , we have that y−β fcosq ≤ Cxγfp, 1 < p ≤ q < ∞, holds with β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 2 + 1/q, β = 1/q.

• A. Debernardi–CRM

WNI’s for integral transforms 19 / 22

Application

As a simple example of the latter, consider f with ∞

0 f(x) dx = 0.

Since cos xy =

∞

• n=0

(−1)n(xy)2n (2n)! , we have that y−β fcosq ≤ Cxγfp, 1 < p ≤ q < ∞, holds with β = γ + 1/q − 1/p′ and max{1/q − 1/p′, 0} ≤ β < 2 + 1/q, β = 1/q. max

• 0, 1

q − 1 p′

• 1/q

2 + 1/q β

• A. Debernardi–CRM

WNI’s for integral transforms 19 / 22

References

• L. De Carli,

On the Lp-Lq norm of the Hankel transform and related

• perators,
• J. Math. Anal. Appl. 348 (2008), 366–382.
• D. Gorbachev, E. Liflyand and S. Tikhonov,

Weighted Pitt inequalities for integral transforms, to appear in Indiana Univ. Math.

• H. P. Heinig,

Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. 33 (4) (1984), 573–582.

• P. Heywood and P. G. Rooney,

A weighted norm inequality for the Hankel transformation,

• Proc. Roy. Soc. Edinburgh Sect. A 99 (1–2) (1984), 45–50.
• A. Debernardi–CRM

WNI’s for integral transforms 20 / 22

References

• W. B. Jurkat and G. Sampson

On rearrangement and weight inequalities for the Fourier transform, Indiana Univ. Math. J. 33 (2) (1984), 257–270.

• B. Muckenhoupt,

Weighted norm inequalities for the Fourier transform,

• Trans. Amer. Math. Soc. 276 (2) (1983), 729–742.
• P. G. Rooney,

On the Yν and Hν transformations,

• Canad. J. Math. 32 (5) (1980), 1021–1044.
• C. Sadosky and R. L. Wheeden,

Some weighted norm inequalities for the Fourier transform of functions with vanishing moments,

• Trans. Amer. Math. Soc. 300 (2) (1987), 521–533.
• A. Debernardi–CRM

WNI’s for integral transforms 21 / 22

Thank you!