Introduction to Variable Exponent Lebesgue

What is/are Exponent Lebesgue?

Exponent Lebesgue - The reconstruction is obtained by means of a Landweber-like iterative method performing a regularization in the framework of variable-exponent Lebesgue spaces. ^[1] We study the eigenvalue problem for the general Kirchhoff's equation − M ( ∫ Ω | ∇ u ( y ) | p ( y ) d y ) div ( | ∇ u ( x ) | p ( x ) − 2 ∇ u ( x ) ) = λ | u ( x ) | q ( x ) − 2 u ( x ) , for suitable M, in the context of variable exponent Lebesgue spaces. ^[2] The approach is based on a regularization scheme developed in the framework of variable-exponent Lebesgue spaces, which enhances the quality of the reconstruction by properly tuning the exponent function that defines the adopted norm. ^[3] In this article, a multifrequency tomographic approach in nonconstant-exponent Lebesgue spaces is enhanced by a preliminary step that processes the measured scattered field with a neural network based on long short-term memory cells. ^[4] org/1998/Math/MathML" id="M10"> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>p</mi> <mfenced open="(" close=")"> <mrow> <mo>·</mo> </mrow> </mfenced> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </mfenced> </math> </jats:inline-formula> is the variable exponent Lebesgue space. ^[5] In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. ^[6] The acquired scattered-field data are processed by a nonlinear inverse-scattering approach in variable-exponent Lebesgue spaces able to jointly exploit multifrequency data, which is extended here for the first time to subsurface imaging problems. ^[7] In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. ^[8] In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. ^[9] The inverse-scattering scheme is developed in variable-exponent Lebesgue spaces, where a delay-and-sum beamforming approach is preliminary applied to build the exponent function. ^[10] In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. ^[11] Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. ^[12] The aim of this paper is to study the compound Riemann Hilbert boundary value problem in the class of Cauchy-type integral with density in variable exponent Lebesgue spaces. ^[13] Sobolev embeddings for domains with Lipschitz boundaries in $${\mathbb {R}}^n$$ are also derived in the framework of new scales of grand variable exponent Lebesgue spaces. ^[14] In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent Lebesgue spaces. ^[15] We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. ^[16] In this paper, the Kantorovich operators Kn, n ∈ N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0, 1]. ^[17] The flexibility of the methods of proofs allows to prove Leibniz-type rules in a variety of function spaces that include Triebel-Lizorkin and Besov spaces based on weighted Lebesgue, Lorentz and Morrey spaces as well as variable-exponent Lebesgue spaces. ^[18] We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. ^[19] In this paper we recall the weighted vector-valued classical and variable exponent Lebesgue spaces. ^[20] We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)$p(x)$-Laplacian operator with Dirichlet boundary condition: −Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,$$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ where Ω is a smooth bounded domain in RN$\mathbb{R}^{N}$, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t)$f(x,t)$ is a Carathéodory function satisfying some growth conditions. ^[21] Based on some pointwise estimates, we establish the boundedness of multilinear commutators of Marcinkiewicz integrals in variable exponent Lebesgue spaces, which in turn are used to obtain some boundedness results for such operators in variable exponent Herz and HerzMorrey spaces. ^[22] In the present work we investigate the approximation of the functions by the Zygmund means in variable exponent Lebesgue spaces. ^[23] Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. ^[24] Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1≤esinfx∈0,2π]p(x), esssup∈[0,2π]p(x)<∞, and the Dini–Lipschitz condition. ^[25] We consider several fundamental properties of grand variable exponent Lebesgue spaces. ^[26] The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. ^[27] In this paper we characterize those exponents p ( ⋅ ) for which corresponding variable exponent Lebesgue space L p ( ⋅ ) ( [ 0 ; 1 ] ) has in common with L ∞ the property that the space of continuous functions is a closed linear subspace in it. ^[28] We prove optimality results for the action of the operator Aα on variable exponent Lebesgue spaces Lp(·) and weighted variable exponent Lebesgue spaces, as an extension of [13, 14, 17]. ^[29] For the last quarter century a considerable number of research has been carried out on the study of Herz spaces, variable exponent Lebesgue  spaces and Sobolev spaces. ^[30] Moreover, the recent extension to the more general case of variable-exponent Lebesgue spaces is also addressed. ^[31] We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. ^[32] In this paper, a necessary and sufficient condition, such as the Pontryagin’s maximum principle for an optimal control problem with distributed parameters, is given by the third-order Bianchi equation with coefficients from variable exponent Lebesgue spaces. ^[33]

Variable Exponent Lebesgue

We study the eigenvalue problem for the general Kirchhoff's equation − M ( ∫ Ω | ∇ u ( y ) | p ( y ) d y ) div ( | ∇ u ( x ) | p ( x ) − 2 ∇ u ( x ) ) = λ | u ( x ) | q ( x ) − 2 u ( x ) , for suitable M, in the context of variable exponent Lebesgue spaces. ^[1] org/1998/Math/MathML" id="M10"> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>p</mi> <mfenced open="(" close=")"> <mrow> <mo>·</mo> </mrow> </mfenced> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </mfenced> </math> </jats:inline-formula> is the variable exponent Lebesgue space. ^[2] In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. ^[3] In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. ^[4] In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. ^[5] In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. ^[6] Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. ^[7] The aim of this paper is to study the compound Riemann Hilbert boundary value problem in the class of Cauchy-type integral with density in variable exponent Lebesgue spaces. ^[8] Sobolev embeddings for domains with Lipschitz boundaries in $${\mathbb {R}}^n$$ are also derived in the framework of new scales of grand variable exponent Lebesgue spaces. ^[9] In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent Lebesgue spaces. ^[10] We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. ^[11] In this paper, the Kantorovich operators Kn, n ∈ N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0, 1]. ^[12] We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. ^[13] In this paper we recall the weighted vector-valued classical and variable exponent Lebesgue spaces. ^[14] We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)$p(x)$-Laplacian operator with Dirichlet boundary condition: −Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,$$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ where Ω is a smooth bounded domain in RN$\mathbb{R}^{N}$, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t)$f(x,t)$ is a Carathéodory function satisfying some growth conditions. ^[15] Based on some pointwise estimates, we establish the boundedness of multilinear commutators of Marcinkiewicz integrals in variable exponent Lebesgue spaces, which in turn are used to obtain some boundedness results for such operators in variable exponent Herz and HerzMorrey spaces. ^[16] In the present work we investigate the approximation of the functions by the Zygmund means in variable exponent Lebesgue spaces. ^[17] Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. ^[18] Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1≤esinfx∈0,2π]p(x), esssup∈[0,2π]p(x)<∞, and the Dini–Lipschitz condition. ^[19] We consider several fundamental properties of grand variable exponent Lebesgue spaces. ^[20] The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. ^[21] In this paper we characterize those exponents p ( ⋅ ) for which corresponding variable exponent Lebesgue space L p ( ⋅ ) ( [ 0 ; 1 ] ) has in common with L ∞ the property that the space of continuous functions is a closed linear subspace in it. ^[22] We prove optimality results for the action of the operator Aα on variable exponent Lebesgue spaces Lp(·) and weighted variable exponent Lebesgue spaces, as an extension of [13, 14, 17]. ^[23] For the last quarter century a considerable number of research has been carried out on the study of Herz spaces, variable exponent Lebesgue  spaces and Sobolev spaces. ^[24] We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. ^[25]

exponent lebesgue space

The reconstruction is obtained by means of a Landweber-like iterative method performing a regularization in the framework of variable-exponent Lebesgue spaces. ^[1] We study the eigenvalue problem for the general Kirchhoff's equation − M ( ∫ Ω | ∇ u ( y ) | p ( y ) d y ) div ( | ∇ u ( x ) | p ( x ) − 2 ∇ u ( x ) ) = λ | u ( x ) | q ( x ) − 2 u ( x ) , for suitable M, in the context of variable exponent Lebesgue spaces. ^[2] The approach is based on a regularization scheme developed in the framework of variable-exponent Lebesgue spaces, which enhances the quality of the reconstruction by properly tuning the exponent function that defines the adopted norm. ^[3] In this article, a multifrequency tomographic approach in nonconstant-exponent Lebesgue spaces is enhanced by a preliminary step that processes the measured scattered field with a neural network based on long short-term memory cells. ^[4] org/1998/Math/MathML" id="M10"> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>p</mi> <mfenced open="(" close=")"> <mrow> <mo>·</mo> </mrow> </mfenced> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </mfenced> </math> </jats:inline-formula> is the variable exponent Lebesgue space. ^[5] In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. ^[6] The acquired scattered-field data are processed by a nonlinear inverse-scattering approach in variable-exponent Lebesgue spaces able to jointly exploit multifrequency data, which is extended here for the first time to subsurface imaging problems. ^[7] In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. ^[8] In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. ^[9] The inverse-scattering scheme is developed in variable-exponent Lebesgue spaces, where a delay-and-sum beamforming approach is preliminary applied to build the exponent function. ^[10] In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. ^[11] Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. ^[12] The aim of this paper is to study the compound Riemann Hilbert boundary value problem in the class of Cauchy-type integral with density in variable exponent Lebesgue spaces. ^[13] Sobolev embeddings for domains with Lipschitz boundaries in $${\mathbb {R}}^n$$ are also derived in the framework of new scales of grand variable exponent Lebesgue spaces. ^[14] In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent Lebesgue spaces. ^[15] We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. ^[16] In this paper, the Kantorovich operators Kn, n ∈ N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0, 1]. ^[17] The flexibility of the methods of proofs allows to prove Leibniz-type rules in a variety of function spaces that include Triebel-Lizorkin and Besov spaces based on weighted Lebesgue, Lorentz and Morrey spaces as well as variable-exponent Lebesgue spaces. ^[18] We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. ^[19] In this paper we recall the weighted vector-valued classical and variable exponent Lebesgue spaces. ^[20] We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)$p(x)$-Laplacian operator with Dirichlet boundary condition: −Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,$$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ where Ω is a smooth bounded domain in RN$\mathbb{R}^{N}$, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t)$f(x,t)$ is a Carathéodory function satisfying some growth conditions. ^[21] Based on some pointwise estimates, we establish the boundedness of multilinear commutators of Marcinkiewicz integrals in variable exponent Lebesgue spaces, which in turn are used to obtain some boundedness results for such operators in variable exponent Herz and HerzMorrey spaces. ^[22] In the present work we investigate the approximation of the functions by the Zygmund means in variable exponent Lebesgue spaces. ^[23] Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. ^[24] Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1≤esinfx∈0,2π]p(x), esssup∈[0,2π]p(x)<∞, and the Dini–Lipschitz condition. ^[25]