Boundedness of vector-valued B-singular integral operators in Lebesgue spaces

Seyda Keles; Mehriban N. Omarova

doi:10.1515/math-2017-0081

Article Open Access

Boundedness of vector-valued B-singular integral operators in Lebesgue spaces

Seyda Keles and Mehriban N. Omarova

Published/Copyright: July 21, 2017

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$Open Mathematics$

From the journal Open Mathematics Volume 15 Issue 1

Abstract

We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator

△B=∑k=1n−1∂ 2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0.

We prove the boundedness of vector-valued B-singular integral operators A from Lp,v(R+n,H1)toLp,v(R+n,H2), 1 < p < ∞, where H₁ and H₂ are separable Hilbert spaces.

Keywords: Laplace-Bessel differential operator; Generalized shift operator; Vector-valued B-singular integral operators

MSC 2010: 42B20; 44A35; 46E40

1 Introduction

Multidimensional singular integrals generated by Laplace-Bessel differential operator (B-singular integral operator)

△B=∑k=1n−1∂2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0

were investigated by M. Klyuchantsev and I.A. Kipriyanov [7, 8], where weighted L_p estimates were obtained for them. I.A. Aliev, A.D. Gadjiev, E.V. Guliyev, I. Ekincioglu, A. Serbetci [1, 3-5] have studied the boundedness of B-singular integrals in weighted L_p-spaces with radial and general weights consequently.

The structure of the paper is as follows. In section 2, we present some definitions and auxiliary results. In section 3 we prove the boundedness of vector-valued B-singular integral operator A from Lp,v(R+n,H1)toLp,v(R+n,H2), 1 < p < ∞, where H₁ and H₂ are separable Hilbert spaces.

2 Definition, notation and preliminaries

Let ℝⁿ be n- dimensional Euclidean space, R+n={x=(x1 ,⋯,xn)∈R:xn>0},S+n−1={x∈R+n:|x|=1},|x|=(∑i=1nxi2)1 /2. We define the spaces Lp,v(R+n) by

Lp,v(R+n):=Lp(R+n,xn2vdx)={f:‖f‖L p , v( R + n)=(∫ R + n|f(x)|pxn2vdx)1p<∞},

where v > 0 is a fixed parameter, 1 ≤ p < ∞, dx = dx₁dx₂ ... dx_n.

Suppose that 1 ≤ p < ∞ and f is a measurable function on R+n. The weak Lp,v(R+n) spaces are denoted by WLp,v(R+n) and f is said to belong to WLp,v(R+n) , if there is a constant C > 0 such that

supλ>0λ|{x∈R+n:|f(x)|>λ}|v1 p ≤C<∞.

For x∈R+n and r > 0, we denote by E(x,r)={y∈R+n:|x−y|<r} the open ball centered at x of radius r, by ∁E(x,r)=R+n∖E(x,r) denote its complement. For any A⊂R+n,|A|v=∫Axn2vdx and |E(0, r)|_v = Cr^n+2v.

We denote by T^y the generalized shift operator (B- shift operator) acting according to the law

Tyf(x)=Γ(v+12)Γ(v)Γ(12)∫0πf(x′−y′,xn2−2xnyncos⁡αn+yn2)(sin⁡α)2v−1dα,

where x = (x′, x_n), y = (y′, y_n), x′, y′ ∈ ℝⁿ⁻¹ and x_n, y_n ∈ ℝ⁺. We remark that T^y is closely connected with the Laplace- Bessel differential operator

ΔB=∑k=1n−1∂2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0

(see [9] for details). The T^y shift operator generates the corresponding “B-convolution”

(φ⊗ψ)(x)=∫ R + nφ(y)Tyψ(x)yn2vdy

for which the Young inequality holds:

‖φ⊗ψ‖L r , v( R + n) ≤‖φ‖L p , v( R + n)‖ψ‖L q , v( R + n),1 ≤p,q,r ≤∞,1p+1q=1r+1.

We note the following property (needed below) of the “B-convolution”,

φ⊗ψ=ψ⊗φ

is valid.

The Fourier-Bessel transform is defined as follows [6]

Fvφ(z)=∫ R + nφ(x)e−ix′z′jv−12(xnzn)xn2vdx,

where jv−12 is the normalized Bessel function that is defined as for t>0,v>−12

jv(t)=2vΓ(v+1)J v(t)tv=Γ(v+1)πΓ(v+12)∫−11(1−u2)v−12eitudu.

Here J_v(t) is the first kind of Bessel function.

The normalized Bessel function j_v is the eigenfunction of the Bessel differential operator satisfying the conditions for all t ∈ ℝ, |j_v(t)| ≤ 1, j_v(0) = 1, jv′(0)=0 and for λ ∈ ℂ

jv(λx)jv(λy)=Tx(jv(λ⋅))(y).

The influence of the Fourier- Bessel transform to B- convolution is as follows

Fv(φ⊗ψ)(x)=(Fvφ)(x)(Fvψ)(x).

Definition 2.1

Suppose that T is a sublinear operator and 1 ≤ p,q ≤ ∞. The operator T is said to be of weak type (p, q)_v, if T is a bounded operator from Lp,v(R+n)toWLq,v(R+n). That is, there exists a constant C > 0 such that for any λ > 0 and f∈Lp,v(R+n)

|{x∈R+n:|Tf(x)|>λ}|v ≤(Cλ‖f‖L p , v( R + n))q .

T is said to be of type (p, q)_v if T is a bounded operator from Lp,v(R+n)toLq,v(R+n). That is, there exists a constant C > 0 such that for any f∈Lp,v(R+n)

‖Tf‖L q , v( R + n) ≤C‖f‖L p , v( R + n).

Theorem 2.2

(The Marcinkiewicz interpolation theorem). Let (X, μ) and (Y, v) be two measure spaces, and let the sublinear operator T be both of weak type (p₀, p₀) and weak type (p₁, p₁) for 1 ≤ p₀ < p₁ ≤ ∞. That is, there exist the constants C₀, C₁ > 0 such that for any λ > 0 and f

(v ( { y ∈ Y : | T f ( y ) | > λ } ) ) 1 p 0 < _ C 0 λ ‖ f ‖ L p 0 ( X , μ ) , (v({y∈Y: |Tf(y) |>λ} ) ) 1 p 1 ≤ C 1 λ ‖ f ‖ L p 1 ( X , μ )

for p₁ < ∞. Then T is also type (p, p) for all p₀ < p < p₁ i.e., there exists a constant C > 0 such that for any f ∈ L_p (X, μ)

(∫Y|Tf(y)|pdv)1p ≤C(∫X|f(x)|pdμ)1p .

3 Vector-valued B-singular integral operators

Let H be a separable Hilbert space. Then a function f(x), from R+n to H is measurable if the scalar valued functions 〈f(x), ϕ〉 are measurable, where 〈·, ·〉 denotes the inner product of H and ϕ denotes an arbitrary vector of H. If f(x) is such a measurable function, then ||f(x)||_H is also measurable (as a function with non-negative values), where ||·||_H denotes the norm of H. Thus Lp,v(R+n,H) is defined as the equivalence classes of measurable functions f(x) from R+n to H, with the property that the norm

‖f‖L p , v( R + n,H)=(∫ R + n‖f(x)‖Hpxn2vdx)1p,1 ≤p<∞

is finite. When p = ∞ there is a similar definition.

‖ f ‖ L ∞ ( R + n , H ) = e s s sup x ∈ R + n ‖ f ( x ) ‖ H .

Now, let H₁ and H₂ be two separable Hilbert spaces, and let B(H₁, H₂) be the Banach space of bounded linear operators from H₁ to H₂, with the usual operator norm. We say that a function f(x), from R+n to B(H₁, H₂) is measurable if f(x)ϕ is an H₂— valued measurable function for every ϕ ∈ H₁. In this case ||f(x)||_{B(H₁, H₂)} is also measurable and we can define the space Lp,v(R+n,B(H1,H2)). The usual facts about B-convolution hold in this setting. For example, suppose g∈Lq,v(R+n,B(H1,H2))andf∈Lp,v(R+n,B(H1,H2)),1 ≤p,q ≤∞. Then

(f⊗g)(x)=∫ R + ng(y)Tyf(x)yn2vdy

converges in the norm of H₂ almost every x, and

‖(f⊗g)(x)‖H 2 ≤∫ R + n‖g(y)‖B(H 1,H 2)‖Tyf(x)‖H 1yn2vdy.

Also, when 1r=1p+1q−1 with 1 ≤ r ≤ ∞,

‖f⊗g‖L r , v( R + n H 2 ) ≤ ‖g‖L q , v( R + n,B(H 1,H 2))‖f‖L p , v( R + n,H 1).

Now, we suppose that f∈L1,v(R+n,H). Then we can define its H-valued Fourier-Bessel transform

(Fvf)(x)=∫ R + nf(y)e−ix′y′jv−12(xnyn)yn2vdy,

which is an element of L∞(R+n,H).Iff∈L1,v(R+n,H)∩L2,v(R+n,H),thenFvf∈L2,v(R+n,H). The Fourier- Bessel transform can then be extended by continuity to a unitary mapping of the Hilbert space L2,v(R+n,H) to itself (see for details [10]; p.45-46, [11]; p. 307-309).

Theorem 3.1

Let A be a sublinear operator defined on L∞0(R+n,H1) i.e., compactly supported, bounded H₁ valued functions, with values in M(R+n,H2) i.e., the space of measurable, H₂ valued functions. Suppose in addition that for f in L∞0(R+n,H1)

|{x:‖Af‖H 2>λ}|v ≤c1,vλ‖f‖L 1 , v( R + n,H 1)

and

(|{x:‖Af‖H 2>λ}|v)1r ≤cr,vλ‖f‖L r , v( R + n,H 1),

where c_1,v and c_r,v are independent of λ and f. Then for each 1 < p < r, we have that Af∈Lp,v(R+n,H2) whenever f∈Lp,v(R+n,H1) and there is a constant c = c_1,r,p,v independent of f such that

‖Af‖L p , v( R + n,H 2) ≤c‖f‖L p , v( R + n,H 1).

Proof

Let

F(x)={ f ( x ) ‖ f ( x ) | | H 1 , f ( x ) ≠ 0 0 , f ( x ) = 0.

For a scalar valued function g consider Bg(x) = ||A(F(x)g)||_H₂. Clearly B is a sublinear mapping, simultaneously of weak types (1, 1)_v and (r, r)_v with norm ≤ c_1,v,c_r,v, respectively. By the Theorem 2.2 there is a constant c as indicated above so that

‖Bg‖L p , v( R + n) ≤c‖g‖L p , v( R + n),1<p<r.

Upon setting g(x) = ||f(x)||_H₁ our proof is complete. □

Definition 3.2

We say that a function K on R+n whose values are bounded operators from H₁ to H₂ is a vector-valued B-singular integral kernel provided that

K is measurable and integrable on compacts sets not containing the origin,
There exists M > 0 for all ɛ : 0 < ɛ < r
‖∫E(0,r)∖E(0,ϵ)K(x)xn2vdx‖B(H 1,H 2) ≤M
and for each h in H₁,
[∫E(0,r)∖E(0,ϵ)K(x)xn2vdx]h
converges as ɛ → 0,
For each h in H₁ with ||h||_H₁ ≤ 1
∫E(0,4r)∖E(0,r)|x|‖K(x)h‖H 2xn2vdx ≤M,
∫ ∁ E ( 0 , 4 | y | )‖TyK(x)−K(x)‖B(H 1,H 2)xn2vdx ≤M,|y|<14.

Theorem 3.3

Let K : R+n → B(H₁, H₂) be a vector-valued B-singular integral kernel. Then there exists C_v > 0 such that

‖(FvK)(x)‖B(H 1,H 2) ≤CvM,x∈S+n−1 .

Proof

Let K_t(x) := t^n+2vK(tx), t > 0. Then, K_t is the vector-valued B-singular integral kernel. First, observe that

(FvK)(x)=∫ R + n K(y)e−i x ′ y ′jv− 1 2(xnyn)yn2vdy. (1)

Let us write ty instead of y, then we get

(FvK)(x)=∫ R + nK(ty)e−itx′y′jv−12(txnyn)yn2vtn+2vdy=(FvKt)(tx).

Therefore,

∫ R + n e − i x ′ y ′ j v − 1 2 ( x n y n ) T x K(y) y n 2 v dy= ∫ R + n T x ( e − i x ′ y ′ j v − 1 2 ( x n y n ))K(y) y n 2 v dy= ∫ R + n e − i x ′ ( y ′ − x ′ ) T x n ( j v − 1 2 ( x n y n ))K(y) y n 2 v dy= ∫ R + n e − i x ′ y ′ e i | x ′ | 2 j v − 1 2 ( x n y n ) j v − 1 2 ( x n 2 )K(y) y n 2 v dy= e i | x ′ | 2 j v − 1 2 ( x n 2 ) ∫ R + n e − i x ′ y ′ j v − 1 2 ( x n y n )K(y) y n 2 v dy= e i | x ′ | 2 j v − 1 2 ( x n 2 )( F v K)(x). (2)

From (1) and (2), we have

( e i | x ′ | 2 j v − 1 2 ( x n 2 )−1)( F v K)(x)= ∫ R + n e − i x ′ y ′ j v − 1 2 ( x n y n ) ( T x K ( y ) − K ( y ) ) y n 2 v d y= ∫ ∁ E ( 0 , 4 ) e − i x ′ y ′ j v − 1 2 ( x n y n )( T x K(y)−K(y)) y n 2 v dy+ ∫ E ( 0 , 4 ) e − i x ′ y ′ j v − 1 2 ( x n y n )( T X K(y)−K(y)) y n 2 v dy= I 3 (x)+ I 4 (x).

Then for all x∈S+n−1, , we have

‖I3(x)‖B(H 1,H 2) ≤∫∁ E ( 0 , 4 )‖TxK(y)−K(y)‖B(H 1,H 2)yn2vdy ≤M.

Now, I₄(x) requires some work. We rewrite it as

I 4 (x)= ∫ E ( 0 , 4 ) e − i x ′ y ′ j v − 1 2 ( x n y n )( T x K(y)−K(y)) y n 2 v dy= ∫ E ( 0 , 4 ) ( e − i x ′ y ′ j v − 1 2 ( x n y n ) − 1 ) T x K ( y ) y n 2 v d y− ∫ E ( 0 , 4 ) ( e − i x ′ y ′ j v − 1 2 ( x n y n )−1)K(y) y n 2 v dy− ∫ E ( 0 , 4 ) K(y) y n 2 v dy+ ∫ E ( 0 , 4 ) T x K(y) y n 2 v dy= L 1 (x)+ L 2 (x)+ L 3 + L 4 (x).

The following inequality is valid for all x∈S+n−1,

| e − i x ′ y ′ j v − 1 2 ( x n y n )−1 |≤ | j v − 1 2 ( x n y n ) | | e − i x ′ y ′ −1 |+ | j v − 1 2 ( x n y n )−1 | ≤ | e − i x ′ y ′ −1 |+ c v ∫ − 1 1 | e − i x n y n t −1 | |1− t 2 | v − 1 dt ≤C |y |.

Then

‖ L 2 (x) ‖ B ( H 1 , H 2 ) ≤ ∫ E ( 0 , 4 ) | e − i x ′ y ′ j v − 1 2 ( x n y n )−1 |‖K(y) ‖ B ( H 1 , H 2 ) y n 2 v dy≤C ∫ E ( 0 , 4 ) |y |‖K(y) ‖ B ( H 1 , H 2 ) y n 2 v dy≤ ∑ k = 0 ∞ ∫ 4 − k < | y | < 4 − k + 1 |y |‖K(y) ‖ B ( H 1 , H 2 ) y n 2 v dy ≤ M ∑ k = 0 ∞ 4 − k ≤ M . (3)

Since the operator K is a vector-valued B-singular integral kernel, then ∃ M > 0, for all ɛ > 0, such that

‖∫E(0,4)∖E(0,ϵ)K(y)yn2vdy‖B(H 1,H 2)≤M

and

‖∫E(0,4)K(y)yn2vdy‖B(H 1,H 2)=‖limϵ→0+∫E(0,4)∖E(0,ϵ)K(y)yn2vdy‖B(H 1,H 2)≤M.

Therefore,

‖L3(x)‖B( H 1 , H 2 )=‖∫E(0,4)K(y)yn2vdy‖B( H 1 , H 2 )≤M. (4)

Now, we estimate L₁(x).

L1(x)=∫ R + nTx[χE(0,4)(y)(e−ix′y′jv−12(xnyn)−1)]K(y)yn2vdy.

Note that,

| T x χ E ( 0 , 4 ) (y) |≤ c V ∫ 0 π | χ E ( 0 , 4 ) ( x ′− y ′, x n 2 − 2 x n y n cos ⁡ α + y n 2 ) |(sin⁡α ) 2 v − 1 dα≤1.

For α ∈ (0, π) if

|(x′−y′,xn2−2xnyncos⁡α+yn2)|≥4,

then

TxχE(0,4)(y)=0.

We denote

Dx={y∈R+n:|(x′−y′,x n 2−2x ny ncos⁡α+y n 2|<4forα∈(0,π)}.

We have

‖ L 1 (x) ‖ B ( H 1 , H 2 ) = ‖ ∫ D x T x ( e − i x ′ y ′ j v − 1 2 ( x n y n )−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) = ‖ ∫ D x ( e − i x ′ ( y ′ − x ′ ) ( T x n j v − 1 2 ( x n y n ))−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) = ‖ ∫ D x ( e − i x ′ y ′ e i | x ′ | 2 j v − 1 2 ( x n y n ) j v − 1 2 ( x n 2 )−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) ≤ ‖ ∫ D x ( e i | x ′ | 2 j v − 1 2 ( x n 2 ))( e − i x ′ y ′ j v − 1 2 ( x n y n )−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) + ‖ ∫ D x ( e i | x ′ | 2 j v − 1 2 ( x n 2 )−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) = L 5 (x)+ L 6 (x).

We note that

|y| ≤|y−x|+|x|≤|(x′−y′,x n 2−2x ny ncos⁡α+y n 2)|+1<5.

Hence, for all x∈S+n−1 similarly (3) we have

We show that L₆(x) is bounded. Indeed

L 6 (x)= ‖ ∫ D x ( e i | x ′ | 2 j v − 1 2 ( x n 2 )−1)K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) ≤ ‖ ∫ D x K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) ≤ ‖ ∫ E ( 0 , 5 ) K ( y ) y n 2 v d y ‖ B ( H 1 , H 2 ) ≤ M . (6)

Using (5) and (6) , we have

‖L1(x)‖B(H 1,H 2)≤M, (7)

where the constant M > 0 is independent of x∈S+n−1. Finally, we estimate L₄(x).

L 4 (x)= ∫ E ( 0 , 4 ) T x K(y) y n 2 v dy= ∫ R + n T x K(y)( χ E(0,4) (y)) y n 2 v dy= ∫ R + n K(y) T x ( χ E ( 0 , 4 ) (y)) y n 2 v dy= ∫ R + n ∖ Z K(y) T x ( χ E(0,4) (y)) y n 2 v dy,

where

Z={y∈R+n:α∈(0,π),|(x′−y′,x n 2−2x ny ncos⁡α+y n 2)|≥4}.

For y∈R+n∖Zandx∈S+n−1, the following inequalities are valid.

|y|≤|y−x|+|x|≤5

and

|(x′−y′,xn2−2xnyncos⁡α+yn2)|≤|x−y~|≤|x|+|y|

where ỹ = (y′, —y_n) and |ỹ| = |y|. We have

L 4 (x)= ∫ R + n ∖ Z K(y) T x ( χ E ( 0 , 4 ) (y)) y n 2 v dy= ∫ E ( 0 , 3 / 2 ) K(y) T x ( χ E ( 0 , 4 ) (y)) y n 2 v dy+ ∫ ∁ E ( 0 , 3 / 2 ) K(y) T x ( χ E ( 0 , 4 ) (y)) y n 2 v dy.

For E(0, 3/2), |(x′−y′,xn2−2xnyncos⁡α+yn2)|<|x|+|y|<3 and we get

∫E(0,3/2)K(y)Txχ E(0,4) (y))yn2vdy=∫E(0,3/2)K(y)yn2vdy.

Hence

‖ L 4 (x) ‖ B ( H 1 , H 2 ) ≤ ‖ ∫ E ( 0 , 3 / 2 ) K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) + ‖ ∫ E ( 0 , 5 ) ∖ E ( 0 , 3 / 2 ) K(y) y n 2 v dy ‖ B ( H 1 , H 2 ) ≤M. (8)

Using (3), (4), (7) and (8) , we have a constant M > 0 for all x∈S+n−1 such that the inequality

‖I4(x)‖B(H 1,H 2)≤M

is valid.

Therefore the proof is complete. Indeed

‖(FvK)(x)‖B(H 1,H 2)≤Mβ−1,

where

0<β:=|ei|x′ | 2jv−12(xn2)−1|≤2.

□

From Theorem 3.3 we get the following theorem.

Theorem 3.4

Suppose that K : R+n → B(H₁, H₂) is a vector-valued B-singular integral kernel and f∈L∞0(R+n,H1). Let

(Tϵf)(x)=∫ ∁ E ( 0 , ϵ )K(y)Tyf(x)yn2vdy, ϵ>0.

Then there exists a constant C_v > 0 such that for all f∈L2,v(R+n) the inequality

‖Tϵf‖L 2 , v( R + n,H 2)≤Cv‖f‖L 2 , v( R + n,H 1)

is valid.

Proof

Let K_ɛ(x) = K(x)χc_E(0,ɛ)(x). Then we can write (T_ɛf)(x) = (K_ɛ ⊗ f)(x). Using Parseval-Plancherel identity and Theorem 3.3 we have

‖ T ϵ f ‖ L 2 , v ( R + n , H 2 ) =‖ K ϵ ⊗f ‖ L 2 , v ( R + n , H 2 ) =‖( F v K ϵ )( F v f) ‖ L 2 , v ( R + n , H 2 ) = ( ∫ R + n ‖ F v K ϵ (x) ‖ B ( H 1 , H 2 ) 2 ‖ F v f(x) ‖ H 1 2 x n 2 v dx ) 1 2 = ( ∫ s + ∫ ϵ ∞ ‖ F v K(rθ) ‖ B ( H 1 , H 2 ) 2 ‖ F v f(rθ) ‖ H 1 2 r n + 2 v − 1 θ n 2 v drdθ ) 1 2 ≤ C v ( ∫ ∁ E ( 0 , ϵ ) ‖ F v f(x) ‖ H 1 2 x n 2 v dx ) 1 2 ≤ C v ‖f ‖ L 2 , v ( R + n , H 1 ) .

□

The scalar valued case of the following theorem was proved in [10] and [11]. In the vector-valued case the proof is the same.

Theorem 3.5

Let H be a separable Hilbert space, f∈L1,v(R+n,H) and t be a positive constant. Then there exists a decomposition of R+n

R+n=F+∪Ω+, F+∩Ω+=∅

such that

||f(x)||_H ≤ t, a.e x in F⁺,
Ω+=⋃i∈NΩi+,
where Ω i + = { x ∈ R + n : | x − x i | < b i } , i ∈ N a n d Ω i + ∩ Ω j + = ∅ , f o r i ≠ j . Moreover, if
v(x)={ 1 | Ω i + | ∫ Ω i + f ( x ) x n 2 v d x , x ∈ Ω i + , f ( x ) , x ∈ F + ,
then
t≤‖v(x)‖H≤2n+2vt, x∈Ωi+,
f(x)=v(x)+∑iωi(x),whereωi∈L1,v(R+n,H),∫ R + n ωi(x)xn2vdx=0andωi(x)=0forx∉Ωi+,
‖v‖ L 1 , v ( R + n ,H)+∑i‖ωi‖ L 1 , v ( R + n ,H)≤C‖f‖ L 1 , v ( R + n ,H),
|Ω+|v≤t−1‖f‖L 1 , v( R + n,H).

The following theorem is valid.

Theorem 3.6

Let K∈L1,v(R+n,B(H1,H2)) and Af(x) = (K ⊗ f)(x). Suppose that there exist constants B, M > 0 such that for all y ∈ E( 0, 1/B)

∫ ∁ E ( 0 , B ) ‖TyK(x)−K(x)‖B( H 1 , H 2 )xn2vdx≤M (9)

and there exists a constant C_v > 0 such that for all f∈L2,v(R+n,H1) the following inequality

∫ R + n ‖Af(x)‖ H 2 2xn2vdx≤Cv∫ R + n ‖f(x)‖ H 1 2xn2vdx (10)

is valid. Then the inequality

|{x∈R+n:‖Af(x)‖H 2>λ}|v≤Cvλ∫ R + n‖f(x)‖H 1xn2vdx

is valid, where the constant C_v > 0 is independent of f.

Proof

For any s > 0 and f∈L1,v(R+n,H1) that has compact support, we use Theorem 3.5. Hence

| {x∈ R + n :‖Af(x) ‖ H 2 >t} | v ≤ |{x∈ R + n :‖Av(x) ‖ H 2 > t 2 } | v + | {x∈ R + n :‖Aω(x) ‖ H 2 > t 2 } | v

and

‖ v ‖ L 2 , v ( R + n , H 1 ) 2 = ∫ F + ‖ v ( x ) ‖ H 1 2 x n 2 v d x + ∫ Ω + ‖ v ( x ) ‖ H 1 2 x n 2 v d x ≤ C v 2 s ‖ f ‖ L 1 , v ( R + n , H 1 ) , (11)

where f=v+ω,ω=∑i=1∞ωi. From (10) and (11) we have

‖Av‖L 2 , v( R + n,H 2)≤Cv‖v‖L 2 , v( R + n,H 1)≤Cvs12‖f‖ L 1 , v( R + n,H 1)12.

Therefore

| {x∈ R + n :‖Av(x) ‖ H 2 > t 2 } | v = 4 t 2 ∫ { x ∈ R + n : | | A v ( x ) | | H 2 > t 2 } ( t 2 ) 2 x n 2 v dx≤ 4 t 2 ∫ R + n ‖Av(x) ‖ H 2 2 x n 2 v dx≤ 4 C v t 2 ∫ R + n ‖v(x) ‖ H 1 2 x n 2 v dx≤ C v s t 2 ‖f ‖ L 1 , v ( R + n , H 1 ) .

Now, we estimate Aω. Firstly, we consider a ω₁ function. Let

Ω1={x∈Rn:|x|<1B}

such that supp ω1⊂Ω1,Ω1+=Ω1∩R+n and

∫Ω 1 +ω1(x)xn2vdx=0.

Let also Q1={x∈Rn:|x|<B}andQ1+=Q1∩R+n. Then, we have

A ω 1 (x)= ∫ R + n T y K(x) ω 1 (y) y n 2 v dy= ∫ Ω 1 + T y K(x) ω 1 (y) y n 2 v dy= ∫ Ω 1 + [ T y K(x)−K(x)] ω 1 (y) y n 2 v dy.

Using Fubini theorem and the condition (9), we obtain

∫ c Q 1 + ‖A ω 1 (x) ‖ H 2 x n 2 v dx≤ ∫ Ω 1 + ( ∫ c Q 1 + ‖ T y K(x)−K(x) ‖ B ( H 1 , H 2 ) x n 2 v dx)‖ ω 1 (y) ‖ H 1 y n 2 v dy≤M‖ ω 1 ‖ L 1 , v ( R + n , H 1 ) ≤ C v ‖f ‖ L 1 , v ( R + n , H 1 ) .

Using Chebyshev inequality we have

|{x∈ R + n :‖A ω 1 (x) ‖ H 2 >t} | v ≤ | {x∈ R + n : ‖(1− χ Q 1 +(x))A ω 1 (x) ‖ H 2 >t } | v + | Q 1 + | v ≤ C v t‖f ‖ L 1 , v ( R + n , H 1 ) + | Q 1 + | v

and

| Q 1 + | v ≤ C | Ω 1 + | v ≤ C v s ‖ f ‖ L 1 , v ( R + n , H 1 ) .

Therefore,

|{x∈R+n:‖Aω1(x)‖H 2>t}|v ≤Cvt‖f‖L 1 , v( R + n,H 1)+Cvs‖f‖L 1 , v( R + n,H 1)

is obtained. Now, for a ω_i function the support of ω_i is in Ωi={x∈Rn:|x−x(i)|<1B} Then we get

∫ c Q i + | | ∫ Ω i + T y K(x) ω i (y) y n 2 v dy ‖ H 2 x n 2 v dx= ∫ c Q i + | | ∫ Ω i + [ T y K(x)− T x ( i ) K(x)] ω i (y) y n 2 v dy ‖ H 2 x n 2 v dx≤ ∫ Ω i + ‖ ω i (y) ‖ H 1 ( ∫ c Q i + ‖ T y K(x)− T x ( i ) K(x) ‖ B ( H 1 , H 2 ) x n 2 v dx ) y n 2 v dy,

where cQi+={x∈R+n:|x−x(i)|>B}.Ifwewritez=(x′,zn,zn+1)∈R+n+1,zn+1>0, where z_n = x_n cos α, z_n+1 = x_n sin α (0 ≤ α < π), then we get

J=∫cB i +‖K(z′−(x ( i ))′,( z n − x n ( i ) ) 2 + z n + 1 2 )K(z′−y′,( z n − y n ) 2 + z n + 1 2 )−‖B(H 1,H 2)zn+12v−1dz,

where cBi+={z∈R+n+1¯:|zn2+zn+12|>B,|x−x(i)|>B}. By using the change of variable ξ = (ξ′, ξ_n) where ξ′=x′−(x(i))′,ξn=zn−xn(i)andη=(η′,ηn,0)∈R+n+1,whereη′=(x(i))′−y′,ηn=xn(i)−yn, we obtain

J=∫( cB i +)′‖K(ξ′, ξ n 2 + z n + 1 2 )K(ξ′+η′,( ξ n + η n ) 2 + z n + 1 2 )−‖B(H 1,H 2)zn+12v−1dz,

where

(cBi+)′={(ξ′,ξn,zn+1)∈R+n+1:|(ξn+xn(i))2+zn+12−xn(i)|>B,|ξ−xn(i)|>B}.

By using the change of variable ξ_n = x_n cos α, z_n+1 = x_n sin α, x_n > 0, we get

B<|(ξn+xn(i)2+zn+12−xn(i)|≤xn,

and

‖J‖B(H 1,H 2) ≤∫ ∁ E ( 0 , B )‖TηK(x)−K(x)‖B(H 1,H 2)xn2vdx.

Since |η|<1B, then we obtain ||J||_{B(H₁, H₂)} ≤ M and then

∫cQ i +‖Aωi(x)‖H 2xn2vdx≤Cv‖ωi‖L 1 , v( R + n,H 1).

Therefore

|{x∈ R + n :‖Af(x) ‖ H 2 >t} | v ≤ C v ( s t 2 ‖f ‖ L 1 , v ( R + n , H 1 ) + t − 1 ‖f ‖ L 1 , v ( R + n , H 1 ) + s − 1 ‖f ‖ L 1 , v ( R + n , H 1 ) ).

Choosing s > 0 to minimize right hand of this inequality we obtain

|{x∈R+n:‖Af(x)‖H 2>t}|v≤Cvt‖f‖L 1 , v( R + n,H 1).

The proof is completed. □

From Theorem 3.6 we have the following theorem.

Theorem 3.7

Let K∈L1,vloc(R+n,B(H1,H2)),Af(x)=(K⊗f)(x). If there exists M > 0 such that for all y ∈ E(0, 1/4)

∫∁ E ( 0 , 4 | y | )‖TyK(x)−K(x)‖B(H 1,H 2)xn2vdx≤M

and

‖Af‖L 2 , v( R + n,H 2)≤Cv‖f‖L 2 , v( R + n,H 1),

where the constant C_v > 0 is independent of f, then for f∈L∞0(R+n,H1)

‖Af‖L p , v( R + n,H 2)≤Cp,v‖f‖L p , v( R + n,H 1),

where the constant C_p,v > 0 is independent of f.

Proof

For 1 < p < 2, using Theorems 2.2 and 3.6 we have

‖Af‖L p , v( R + n,H 2)≤Cp,v‖f‖L p , v( R + n,H 1).

Note that if a function ψ is H- valued locally integrable function, where H is a separable Hilbert space and if

supφ‖∫ R + nφ(x)ψ(x)xn2vdx‖H=C<∞,

where the sup is taken over all continuous ϕ with scalar- valued compact support which verify ‖φ‖L p ′ , v( R + n)<1, then ψ ∈ L p , v ( R + n , H ) a n d ‖ ψ ‖ L p , v ( R + n , H ) = C , 1 p + 1 p ′ = 1. T h e n f o r 2 < p < ∞ , t a k e f ∈ L 1 , v ( R + n , H 1 ) ∩ L p , v ( R + n , H 1 ) and ϕ of the type described above. Since K∈L1,vloc(R+n,B(H1,H2) and because of our choice of f and ϕ, the double integral

I=∫ R + n∫ R + nTyK(x)f(y)φ(x)yn2vxn2vdxdy

converges absolutely, then

I= ∫ R + n ∫ R + n T y K(x)f(y)φ(x) y n 2 v x n 2 v dxdy= ∫ R + n ( ∫ R + n T y K ( x ) φ ( x ) x n 2 v d x ) f(y) y n 2 v dy.

But if the theorem is valid for 1 < p′ < 2, then

∫ R+nTyK(x)φ(x)xn2vdx

belongs to L p ′,v(R+n,B(H1,H2))anditsL p ′,v(R+n,B(H1,H2)) norm is majorized by

C p ′,v||φ|| L p ′ , v ( R + n)=C p ′,v.

By Hölder’s inequality we show that

||I|| H 2 =‖ ∫ R + n A f ( x ) φ ( x ) x n 2 v d x ‖ H 2 ≤C p ′,v||f|| L p , v ( R + n, H 1 )

and taking the supremum of all the ϕ’s indicated above gives the result that

||Af|| L p , v ( R + n, H 2 )≤C p ′,v||f|| L p , v ( R + n, H 1 ),2<p<∞.

Therefore, we have completed the proof of the theorem. □

The following theorem is our main result.

Theorem 3.8

Let K∈L1,vloc(R+n,B(H1,H2)) be a vector-valued B- singular integral kernel. For f ∈ Lp,v(R+n,H1),1<p<∞, suppose

(Tεf)(x)=∫ ∁ E ( 0 , ε ) K(y)Tyf(x)yn2vdy,ε>0.

Then there exists a constant C_p,v > 0 such that for all f∈Lp,v(R+n,H1) the inequality

||Tεf|| L p , v ( R + n, H 2 )≤Cp,v||f|| L p , v ( R + n, H 1 )

is valid and

Tf(x)=limε→ 0 +(Tεf)(x)

exists in Lp,v(R+n,H2) and also the following inequality is valid

||Tf|| L p , v ( R + n, H 2 )≤Cp,v||f|| L p , v ( R + n, H 1 ) ,

where the constant C_p,v > 0 is independent of f.

Proof

From Theorems 3.4 and 3.7, it follows that

||Tεf|| L p , v ( R + n, H 2 )≤Cp,v||f|| L p , v ( R + n, H 1 ) ,

where the constant C_p,v > 0 is independent of f. Now, we show that Tf(x)=limε→ 0 +(Tεf)(x) exists in Lp,v(R+n,H2) and the operator T so defined also satisfies

||Tf|| L p , v ( R + n, H 2 )≤Cp,v||f|| L p , v ( R + n, H 1 ) ,

where the constant C_p,v > 0 is independent of f. For an arbitrary f∈Lp,v(R+n,H1), we can write

f=f1+f2,

where f₁ is a smooth function with values in H₁ and ||f2|| L p , v ( R+n, H 1) is small. We can write

(Tεf)(x)=(Tεf1)(x)+(Tεf2)(x).

For f2∈Lp,v(R+n,H1) we obtain that

||Tεf2|| L p , v ( R + n, H 2 )≤Cp,v||f2|| L p , v ( R + n, H 1 )<δ,δ>0.

We prove this theorem for a smooth function with values in H₁. For x, y ∈ R+n , we can write

T ε 2 f(x)− T ε 1 f(x)= ∫ E ( 0 , ε 2 ) ∖ E ( 0 , ε 1 ) K ( y ) T y f ( x ) y n 2 v d y= ∫ E ( 0 , ε 2 ) ∖ E ( 0 , ε 1 ) K ( y )( T y f(x)−f(x)) y n 2 v dy+ ∫ E ( 0 , ε 2 ) ∖ E ( 0 , ε 1 ) K ( y ) f ( x ) y n 2 v dy.

By using

||Tyf(⋅)−f(⋅)|| L p , v ( R + n, H 1 )≤C|y|,

we obtain

||T ε 2 f−T ε 1 f|| L p , v ( R + n, H 2 )→0,ε1,ε2→0.

Since Lp,v(R+n,H2) is Banach space and the set of smooth functions with values in H₁ is dense in Lp,v(R+n,H1)thenthelimε→ 0 +(Tεf)(x) exists and hence

||Tf|| L p , v ( R + n, H 2 )≤Cp,v||f|| L p , v ( R + n, H 1 )

is valid, where the constant C_p,v > 0 is independent of f. □

Competing interests

The authors declare that they have no competing interests.
Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgement

The research of M. Omarova are partially supported by the grant of Presidium of Azerbaijan National Academy of Science 2015.

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Received: 2016-12-30

Accepted: 2017-5-8

Published Online: 2017-7-21

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