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The minimal operator and the John--Nirenberg theorem for weighted grand Lebesgue spaces

Lihua Peng, Yong Jiao (2015)

Studia Mathematica

We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt A p class.

The Stein-Weiss Type Inequality for Fractional Integrals, Associated with the Laplace-Bessel Differential Operator

Gadjiev, Akif, Guliyev, Vagif (2008)

Fractional Calculus and Applied Analysis

2000 Math. Subject Classification: Primary 42B20, 42B25, 42B35In this paper we study the Riesz potentials (B -Riesz potentials) generated by the Laplace-Bessel differential operator ∆B.* Akif Gadjiev’s research is partially supported by the grant of INTAS (project 06-1000017-8792) and Vagif Guliyev’s research is partially supported by the grant of the Azerbaijan–U.S. Bilateral Grants Program II (project ANSF Award / 16071) and by the grant of INTAS (project 05-1000008-8157).

The trilinear embedding theorem

Hitoshi Tanaka (2015)

Studia Mathematica

Let σ i , i = 1,2,3, denote positive Borel measures on ℝⁿ, let denote the usual collection of dyadic cubes in ℝⁿ and let K: → [0,∞) be a map. We give a characterization of a trilinear embedding theorem, that is, of the inequality Q K ( Q ) i = 1 3 | Q f i d σ i | C i = 1 3 | | f i | | L p i ( d σ i ) in terms of a discrete Wolff potential and Sawyer’s checking condition, when 1 < p₁,p₂,p₃ < ∞ and 1/p₁ + 1/p₂ + 1/p₃ ≥ 1.

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

Let L be a non-negative self-adjoint operator acting on L 2 ( n ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A r weight on n × n , 1 < r < . In this article we obtain a weighted atomic decomposition for the weighted Hardy space H L , w p ( n × n ) , 0 < p 1 associated to L . Based on the atomic decomposition, we show the dual relationship between H L , w 1 ( n × n ) and BMO L , w ( n × n ) .

Traces of functions with a dominating mixed derivative in 3

Jan Vybíral, Winfried Sickel (2007)

Czechoslovak Mathematical Journal

We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in 3 , with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2 . The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.

Tractable embeddings of Besov spaces into Zygmund spaces

Hans Triebel (2011)

Banach Center Publications

The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces. This can be expressed in terms of tractability envelopes.

Translation averages of dyadic weights are not always good weights.

Lesley A. Ward (2002)

Revista Matemática Iberoamericana

The process of translation averaging is known to improve dyadic BMO to the space BMO of functions of bounded mean oscillation, in the sense that the translation average of a family of dyadic BMO functions is necessarily a BMO function. The present work investigates the effect of translation averaging in other dyadic settings. We show that translation averages of dyadic doubling measures need not be doubling measures, translation averages of dyadic Muckenhoupt weights need not be Muckenhoupt weights,...

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

Suppose that μ is a Radon measure on d , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces p q s ( μ ) for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...

Variable Lebesgue norm estimates for BMO functions

Mitsuo Izuki, Yoshihiro Sawano (2012)

Czechoslovak Mathematical Journal

In this paper, we are going to characterize the space BMO ( n ) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO ( n ) by using various function spaces. For example, Ho obtained a characterization of BMO ( n ) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known...

Wavelet frames for distributions; local and pointwise regularity

Hans Triebel (2003)

Studia Mathematica

This paper deals with wavelet frames for a large class of distributions on euclidean n-space, including all compactly supported distributions. These representations characterize the global, local, and pointwise regularity of the distribution considered.

Wavelet transform for functions with values in UMD spaces

Cornelia Kaiser, Lutz Weis (2008)

Studia Mathematica

We extend the classical theory of the continuous and discrete wavelet transform to functions with values in UMD spaces. As a by-product we obtain equivalent norms on Bochner spaces in terms of g-functions.

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p ( d ) (in the case p > 1 ), but (in the case when 1 / p ( · ) is log-Hölder continuous and p - = inf { p ( x ) : x d } > 1 ) on the variable Lebesgue spaces L p ( · ) ( d ) , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type ( 1 , 1 ) . In the present note we generalize Besicovitch’s covering theorem for the so-called γ -rectangles. We introduce a general maximal operator M s γ , δ and with the help of generalized Φ -functions, the strong- and weak-type...

Weighted bounds for variational Fourier series

Yen Do, Michael Lacey (2012)

Studia Mathematica

For 1 < p < ∞ and for weight w in A p , we show that the r-variation of the Fourier sums of any function f in L p ( w ) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality...

Weighted estimates for commutators of multilinear Hausdorff operators on variable exponent Morrey-Herz type spaces

Dao Van Duong, Kieu Huu Dung, Nguyen Minh Chuong (2020)

Czechoslovak Mathematical Journal

We establish the boundedness for the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belonging to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previously known results.

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