Displaying similar documents to “Boundedness of para-product operators on spaces of homogeneous type”

Boundedness of commutators of strongly singular convolution operators on Herz-type spaces

Zongguang Liu (2003)

Studia Mathematica

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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, T b m , on Herz spaces K ̇ q α , p ( ) and K q α , p ( ) , and on a new class of Herz-type Hardy spaces H K ̇ q , b , m α , p , 0 ( ) and H K q , b , m α , p , 0 ( ) , where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the L p boundedness of the Marcinkiewicz integral operators μ Ω on n × × n k under the condition that Ω L ( l o g L ) k / 2 ( n - 1 × × n k - 1 ) . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Hongbin Wang (2016)

Czechoslovak Mathematical Journal

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Let Ω L s ( S n - 1 ) for s 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μ Ω and b is defined by [ b , μ Ω ] ( f ) ( x ) = ( 0 | x - y | t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] f ( y ) d y | 2 d t t 3 1 / 2 . In this paper, the author proves the ( L p ( · ) ( n ) , L p ( · ) ( n ) ) -boundedness of the Marcinkiewicz integral operator μ Ω and its commutator [ b , μ Ω ] when p ( · ) satisfies some conditions. Moreover, the author obtains the corresponding result about μ Ω and [ b , μ Ω ] on Herz spaces with variable exponent.

On L p - L q boundedness for convolutions with kernels having singularities on a sphere

Alexey N. Karapetyants (2001)

Studia Mathematica

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For the convolution operators A a α with symbols a ( | ξ | ) | ξ | - α e x p i | ξ | , 0 ≤ Re α < n, a ( | ξ | ) L , we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from L p to L q .

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

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Let ( X , d , μ ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ . Let L be a non-negative self-adjoint operator of order m on L 2 ( X ) . Assume that the semigroup e - t L generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H L p ( X ) be the Hardy space associated with L . We show the boundedness of Stein’s square function 𝒢 δ ( L ) arising from Bochner-Riesz means associated to L from Hardy spaces H L p ( X ) to L p ( X ) , and also study...

Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

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Let 1 < p < ∞, q = p/(p-1) and for f L p ( 0 , ) define F ( x ) = ( 1 / x ) ʃ 0 x f ( t ) d t , x > 0. Moser’s Inequality states that there is a constant C p such that s u p a 1 s u p f B p ʃ 0 e x p [ a x q | F ( x ) | q - x ] d x = C p where B p is the unit ball of L p . Moreover, the value a = 1 is sharp. We observe that F = K 1 f where the integral operator K 1 has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...

L p -improving properties of certain singular measures on the Heisenberg group

Pablo Rocha (2022)

Mathematica Bohemica

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Let μ A be the singular measure on the Heisenberg group n supported on the graph of the quadratic function ϕ ( y ) = y t A y , where A is a 2 n × 2 n real symmetric matrix. If det ( 2 A ± J ) 0 , we prove that the operator of convolution by μ A on the right is bounded from L ( 2 n + 2 ) ( 2 n + 1 ) ( n ) to L 2 n + 2 ( n ) . We also study the type set of the measures d ν γ ( y , s ) = η ( y ) | y | - γ d μ A ( y , s ) , for 0 γ < 2 n , where η is a cut-off function around the origin on 2 n . Moreover, for γ = 0 we characterize the type set of ν 0 .

Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations

Dachun Yang (2005)

Studia Mathematica

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Let ( X , ϱ , μ ) d , θ be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, | ϱ ( x , y ) - ϱ ( x ' , y ) | C ϱ ( x , x ' ) θ [ ϱ ( x , y ) + ϱ ( x ' , y ) ] 1 - θ , and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, μ ( y X : ϱ ( x , y ) < r ) r d . Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces F q s ( X ) and establishes their frame characterizations by first establishing a Plancherel-Pólya-type...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .

L p ( ) boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that Ω L ( l o g L ) k + 1 ( S n - 1 ) is a sufficient condition for the kth order commutator to be bounded on L p ( ) for all 1 < p < ∞. The corresponding maximal operator is also considered.

A remark on extrapolation of rearrangement operators on dyadic H s , 0 < s ≤ 1

Stefan Geiss, Paul F. X. Müller, Veronika Pillwein (2005)

Studia Mathematica

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For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator T s , 0 < s < 2, to be the linear extension of the map ( h I ) / ( | I | 1 / s ) ( h τ ( I ) ) ( | τ ( I ) | 1 / s ) , where h I denotes the L -normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that T s is bounded on H s , then for all 0 < s < 2 the operator T s is bounded on H s .

Convolution operators with anisotropically homogeneous measures on 2 n with n-dimensional support

E. Ferreyra, T. Godoy, M. Urciuolo (2002)

Colloquium Mathematicae

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Let α i , β i > 0 , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let t x = ( t α x , . . . , t α x ) , t x = ( t β x , . . . , t β x ) and | | x | | = i = 1 n | x i | 1 / α i . Let φ₁,...,φₙ be real functions in C ( - 0 ) such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on 2 n given by μ ( E ) = χ E ( x , φ ( x ) ) | | x | | γ - α d x , where α = i = 1 n α i and dx denotes the Lebesgue measure on ℝⁿ. Let T μ f = μ f and let | | T μ | | p , q be the operator norm of T μ from L p ( 2 n ) into L q ( 2 n ) , where the L p spaces are taken with respect to the Lebesgue measure. The type set E μ is defined by E μ = ( 1 / p , 1 / q ) : | | T μ | | p , q < , 1 p , q . In the case α i β k for 1 ≤ i,k ≤ n we characterize the...

The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. Ferreyra, T. Godoy (2006)

Colloquium Mathematicae

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Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by μ ( E ) = D χ E ( x , φ ( x ) ) d x with D = x ∈ ℝ ²:|x| ≤ 1 and let T μ be the convolution operator with the measure μ. Let φ = φ e φ e be the decomposition of φ into irreducible factors. We show that if e i m / 2 for each φ i of degree 1, then the type set E μ : = ( 1 / p , 1 / q ) [ 0 , 1 ] × [ 0 , 1 ] : | | T μ | | p , q < can be explicitly described as a closed polygonal region.

The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

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The aim of this paper is to characterize the L p - L q boundedness of two classes of integral operators from L p ( 𝒰 , d V α ) to L q ( 𝒰 , d V β ) in terms of the parameters a , b , c , p , q and α , β , where 𝒰 is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

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We give some rather weak sufficient condition for L p boundedness of the Marcinkiewicz integral operator μ Ω on the product spaces × m (1 < p < ∞), which improves and extends some known results.