Introduction to Lebesgue Space
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Lebesgue Space sentence examples within non negative function
$\end{document} Here \begin{document}$ \Omega $\end{document} is a bounded open subset of \begin{document}$ I\!\!R^{N} (N>p\geq 2), T>0 $\end{document} and \begin{document}$ f $\end{document} is a non-negative function that belong to some Lebesgue space, \begin{document}$ f\in L^{m}(Q) $\end{document} , \begin{document}$ Q = \Omega \times(0,T) $\end{document} , \begin{document}$ \Gamma = \partial\Omega\times(0,T) $\end{document} , \begin{document}$ g(x,t,u) = |u|^{s-1}u $\end{document} , \begin{document}$ s\geq 1, $\end{document} \begin{document}$ 0\leq\theta and \begin{document}$ 0.
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We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt \gamma \lt 1.
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Lebesgue Space sentence examples within | u |
In this article, we show the existence of infinitely many solutions for the fractional p -Laplacian equations of Schröodinger-Kirchhoff type equation $$M\left( {\left[ u \right]_{s,p}^p} \right)( - \Delta )_p^su + V(x){\left| u \right|^{p - 2}}u = \lambda ({I_\alpha }*{\left| u \right|^{p_{s,\alpha }^*}}){\left| u \right|^{p_{s,\alpha }^* - 2}}u + \beta k(x){\left| u \right|^{q - 2}}u,x \in {\mathbb{R}^N},$$ M ( [ u ] s , p p ) ( − Δ ) p s u + V ( x ) | u | p − u = λ ( I α * | u | p s , α * ) | u | p s , α * − u + β k ( x ) | u | q − u , x ∈ R N , where $$(-\Delta )_p^s$$ ( − Δ ) p s is the fractional p -Laplacian operator, [ u ] s , p is the Gagliardo p -seminorm, 0 < s < 1 < q < p < N / s , α ∈ (0, N ), M and V are continuous and positive functions, and k ( x ) is a non-negative function in an appropriate Lebesgue space.
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Lebesgue Space sentence examples within Weighted Lebesgue Space
In a previous paper, we obtained several"compact versions"of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded.
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In the endpoint case, we establish the weighted weak $L\log L$-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces.
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Lebesgue Space sentence examples within 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.
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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.
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Lebesgue Space sentence examples within Grand Lebesgue Space
We introduce a new class of quasi-Banach spaces as an extension of the classical Grand Lebesgue Spaces for ‘‘small’’values of the parameter, and we investigate some its properties, in particular, completeness, fundamental function, operators estimates, Boyd indices, contraction principle, tail behavior, dual space, generalized triangle and quadrilateral constants and inequalities.
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</mml:math></jats:alternatives></jats:inline-formula> condition near the origin, then This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces.
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Lebesgue Space sentence examples within Norm Lebesgue Space
Particularly, when $X:=M_q^p({\mathbb R}^n)$ (the Morrey space), $X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space) and $X:=(E_\Phi^q)_t({\mathbb R}^n)$ (the Orlicz-slice space), which are all ball quasi-Banach function spaces but not quasi-Banach function spaces, all these results are even new.
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In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd).
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Lebesgue Space sentence examples within Suitable Lebesgue Space
We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt \gamma \lt 1.
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Lebesgue Space sentence examples within Anisotropic Lebesgue Space
By considering different weights in spatial variables, we show in anisotropic Lebesgue spaces if ∂ 3 u 3 and b satisfy certain space–time integrable conditions, which are almost optimal from the scaling invariant point of view, then a weak solution ( u , b ) is actually regular.
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In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces.
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Lebesgue Space sentence examples within Mixed Lebesgue Space
Lebesgue Space sentence examples within Variable Lebesgue Space
Particularly, even when X := Lp(·)(Rn) (the variable Lebesgue space), X := L(R) (the mixed-norm Lebesgue space), X := L(R) (the Orlicz space), and X := (E q Φ )t(R ) (the Orlicz-slice space or the generalized amalgam space), all these results are new.
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In this paper we deal with the martingales in variable Lebesgue space over a probability space.
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Lebesgue Space sentence examples within Small Lebesgue Space
Lebesgue Space sentence examples within Weak Lebesgue Space
Lebesgue Space sentence examples within lebesgue space lp
In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space Hp{H_{p}} to the Lebesgue space Lp{L_{p}} for all 0.
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We present an alternative point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space Lp is replaced by the slightly larger Marcinkiewicz space Mp (aka weak Lp space)—a popular tool in harmonic analysis.
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Lebesgue Space sentence examples within lebesgue space filling
We then use the Lebesgue space filling curve to illustrate how a complete arithmetization of a space filling curve can be done on any of its space filling cores.
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Moreover, it is also valid to any other kinds of Space Filling Curves like Lebesgue Space Filling Curve, Peano Curve and Moore Curve.
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In this article, the inversion of S-parameter data collected in a metallic chamber is performed with a nonlinear inversion strategy in Lebesgue spaces with nonconstant exponents.
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This gap can be reduced for data in Fourier-Lebesgue spaces \begin{document}$ \widehat{H}^{s, r} $\end{document} and \begin{document}$ \widehat{H}^{l, r} $\end{document} to \begin{document}$ s> \frac{21}{16} $\end{document} and \begin{document}$ l > \frac{9}{8} $\end{document} for \begin{document}$ r $\end{document} close to \begin{document}$ 1 $\end{document} , whereas the critical exponents with respect to scaling fulfill \begin{document}$ s_c \to 1 $\end{document} , \begin{document}$ l_c \to 1 $\end{document} as \begin{document}$ r \to 1 $\end{document}.
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This paper is devoted to the boundedness and Fredholmness of $$\mathbf{SG}$$
-pseudo-differential operators with non-zero order in the Lebesgue spaces with variable exponent p(x).
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Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.
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A criterion for the maximum possible pointwise convergence rate in Birkhoff’s ergodic theorem for ergodic semiflows in a Lebesgue space is obtained.
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In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes $$W_{q)}^{2} \left( \varOmega \right) $$
on a bounded n-dimensional domain $$\varOmega \subset R^{n} $$
with a sufficiently smooth boundary $$\partial \varOmega $$
, generated by the norm of the grand-Lebesgue space $$L_{q)}\left( \varOmega \right) $$.
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In this paper, on the base of Pontryagin maximum principle, the optimal control problem with concentrated parameters for a degenerate differential equation with the Caputo operator and with coefficients from the Lebesgue space is studied.
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In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces L p (ℝ n ), Hardy spaces H p (ℝ n ) and general mixed norm spaces, which implies almost everywhere convergence of such operator.
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The paper introduces the kth-order slant Toeplitz operator on the Lebesgue space of n-torus, where k = (k1,k2,…,kn) such that kt ≥ 2 for all t.
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In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space Hp{H_{p}} to the Lebesgue space Lp{L_{p}} for all 0.
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On the Lebesgue space of the torus, this paper introduces the class of slant Toeplitz like operators.
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We completely solve a system of operator equations, whose findings are utilized to generalize the notion of slant Toeplitz operators defined on the Lebesgue space of n-dimensional torus 𝕋n.
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We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space.
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We present an alternative point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space Lp is replaced by the slightly larger Marcinkiewicz space Mp (aka weak Lp space)—a popular tool in harmonic analysis.
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We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space $$A^p_{\alpha }$$
into a Lebesgue space $$L^q(\mu )$$
, where $$0< q < p$$
and $$\alpha > -1$$.
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In this paper, we obtain a generalization of Titchmarsh's theorem for the Bessel transform for functions satisfying the $(psi,p)$-Bessel Lipschitz condition in the Lebesgue space $L_{p,gamma}(mathbb{R}^{n}_{+})$ for $1 0$.
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3) maps from the product of local Hardy spaces to the Lebesgue space, i.
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First we establish a local Calderon–Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces L p q , for every q ∈ ( 1 , ∞ ) , provided the partial map ( x , u ) ↦ A ( x , u , ξ ) is Holder continuous and B ( x , u , ξ ) satisfies a suitable growth condition.
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We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces.
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We characterize a two-dimensional bilinear inequality with rectangular Hardy operators in weighted norms of Lebesgue spaces.
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A straightforward transition to normal form is given that applies to the various Hardy operators and their duals, whether defined on Lebesgue spaces of sequences, of functions on the half-line, or of functions on Rn or more general metric spaces.
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The simplest example reads as the following boundary value problem in a bounded domain of $\mathbb{R}^N$ \begin{cases} -div(A(x)\nabla u) + \lambda = H(x, \nabla u) \qquad \hbox{in $\Omega$,} & \\ A(x) \nabla u\cdot \vec n=0\qquad \hbox{on $\partial \Omega$,} & \end{cases} where A(x) is a coercive matrix with bounded coefficients, and $H(x,\nabla u)$ has Lipschitz growth in the gradient and measurable $x$-dependence with suitable growth in some Lebesgue space (typically, $|H(x,\nabla u)|\leq b(x) |\nabla u|+ f(x)$ for functions b(x)∈ LN(Ω) and f (x) ∈ Lm(Ω), $m\geq 1$).
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<jats:p>We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first order, in the Lebesgue space <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p({\mathbb {R}}^d;{\mathbb {R}}^m)$$</jats:tex-math><mml:math xmlns:mml="http://www.
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We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform.
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We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes.
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We establish interior estimates of the form for the elements of this wedge, where is a compact subdomain of , is the Sobolev space, , is the Lebesgue space of integrable functions, and the constant is independent of.
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In particular, a time-difference formulation is adopted, and the resulting ill-posed equation is solved by means of an iterative procedure performing a regularization in the framework of Lebesgue spaces.
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The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance.
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Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.
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To this end, a new hybrid inversion procedure exploiting a qualitative delay-and-sum method together with a conjugate-gradient-like scheme developed in Lebesgue spaces has been used.
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Our aim in this paper is to deal with Frechet-Kolmogorov compactness of Prabhakar integral operator with Prabhakar-like memory kernel on Lebesgue spaces.
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Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.
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An example of a problem in Lebesgue space is given for which the proposed method is applicable.
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We then use the Lebesgue space filling curve to illustrate how a complete arithmetization of a space filling curve can be done on any of its space filling cores.
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Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces Ĥ 1 p , p 2 (R)(4 ≤ p < ∞), Ĥ 3s1 p , 2p 3 (R)(s1 > 1 3 , 3 ≤ p < ∞), Ĥ 2s1 p (R)(s1 > n 2(n+1) , 2 ≤ p < ∞, n ≥ 3) with rough data.
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His method characterizes the $$L^{p}$$
norm in terms of the Lebesgue spaces $$L^{1}$$
and $$L^{\infty }$$
, and works not only for complex Lebesgue spaces but also for real Lebesgue spaces.
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Lebesgue spaces ( L p over R n ) play a significant role in mathematical analysis.
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Furthermore, convergence rates in Lebesgue spaces are obtained also.
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In this paper, we introduce a new version of the definition of a quasi-norm (in particular, a norm) in Lebesgue spaces with variable order of summability.
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