Introduction to Lebesgue Spaces
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Lebesgue Spaces sentence examples within vector valued version
The Kothe–Bochner spaces Lρ(X) are the vector valued version of the scalar Kothe spaces Lρ, which generalize the Lebesgue spaces Lp, the Orlicz spaces and many other functional spaces.
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Lebesgue Spaces sentence examples within Weighted Lebesgue Spaces
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 Spaces sentence examples within Exponent Lebesgue Spaces
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 Spaces sentence examples within Grand Lebesgue Spaces
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 Spaces sentence examples within Variable Lebesgue Spaces
Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p > 1.
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By using the boundedness results for the commutators of the fractional integral with variable kernel on variable Lebesgue spaces Lp(·)(Rn), the boundedness results are established on variable exponent Herz−Morrey spaces MK̇α,λ q,p(·)(R n).
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Lebesgue Spaces sentence examples within Anisotropic Lebesgue Spaces
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 Spaces sentence examples within Norm Lebesgue Spaces
In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd).
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In this paper, we introduce the anisotropic fractional Sobolev spaces and prove that such spaces embed to mixed-norm Lebesgue spaces with certain indices.
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Lebesgue Spaces sentence examples within Mixed Lebesgue Spaces
Lebesgue Spaces sentence examples within Generalized Lebesgue Spaces
Let $$ \mu $$μ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$Rd; $$ p,q,\gamma ,k $$p,q,γ,k real numbers; $$ M_{\mu ,k}^{\gamma } $$Mμ,kγ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$Ap,qγ,kμ a Muckenhoupt condition associated to $$ \mu $$μ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$Lp(·)(Rd,μ) and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$F(q,p,α,μ)(Rd) two generalized Lebesgue spaces.
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\end{aligned}$$Δ(|Δu|p1(x)-2Δu)+Δ(|Δu|p2(x)-2Δu)=λV1(x)|u|q(x)-2u-μV2(x)|u|α(x)-2u,x∈Ωu=Δu=0,x∈∂Ωwhere $$\Omega \in \mathbb {R}^{N}$$Ω∈RN with $$N\ge 2$$N≥2 is a bounded domain with smooth boundary, $$\lambda $$λ, $$\mu $$μ are positive real numbers, $$p_{1}$$p1, $$p_{2}$$p2, q and $$\alpha $$α are continuous functions on $$\overline{\Omega }$$Ω¯, $$V_1$$V1 and $$V_2$$V2 are weight functions in a generalized Lebesgue spaces $$L^{s_1(x)}(\Omega )$$Ls1(x)(Ω) and $$L^{s_2(x)}(\Omega )$$Ls2(x)(Ω) respectively such that $$V_1$$V1 may change sign in $$\Omega $$Ω and $$V_2\ge 0$$V2≥0 on $$\Omega $$Ω.
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Lebesgue Spaces sentence examples within Arbitrary Lebesgue Spaces
To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces.
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To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $$\sigma $$σ-finite measures spaces.
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Lebesgue Spaces sentence examples within Usual Lebesgue Spaces
For the IVP associated with the first one, by using the Orlicz space with the function
$\Xi (z)={\textrm {e}}^{|z|^{p}}-1$
and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular.
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More precisely, if the maximal time of existence of solutions for these equations is finite, we demonstrate the explosion, near this instant, of some limits superior and integrals involving a specific usual Lebesgue spaces and, as a consequence, we prove the lower bounds related to Sobolev–Gevrey spaces.
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Lebesgue Spaces sentence examples within Small Lebesgue Spaces
Lebesgue Spaces sentence examples within Weak Lebesgue Spaces
Lebesgue Spaces sentence examples within lebesgue spaces lp
In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd).
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For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces.
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Lebesgue Spaces sentence examples within lebesgue spaces l
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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We note that the well-known result of Von Neumann [29] is not valid for all doubly substochastic operators on discrete Lebesgue spaces l(I), p ∈ [1,∞).
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Lebesgue Spaces sentence examples within lebesgue spaces defined
To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces.
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This paper is concerned to study the existence and uniqueness of solution of neutral type differential equations, by using the maximal regularity property of the first-order abstract Cauchy problem with finite delay on Lebesgue spaces defined at the line.
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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}$$
-pseudo-differential operators with non-zero order in the Lebesgue spaces with variable exponent p(x).
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Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.
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Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces $$N_{q)}^{2} \left( \varOmega \right) $$
with data from grand-Lebesgue type spaces that are different from Lebesgue spaces.
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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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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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Our results extend to weak solutions and to data in Lebesgue spaces LN(Ω) (or in the dual space (H1(Ω))'), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.
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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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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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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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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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In this paper, we discuss relationships between the continuous embeddings of Dirichlet spaces ( F , E 1 ) into Lebesgue spaces and the integrability of the associated resolvent kernel r α ( x , y ).
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Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.
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This difficulty is overcame by approximating the solution using approximate functions composing of the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously.
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This article addresses two characterizations of BMO(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO(ℝn){\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-closure of Cc∞(ℝn){C_{c}^{\infty}(\mathbb{R}^{n})} space via the compactness of the commutators.
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Under assumption that λ \lambda satisfies ε \varepsilon -weak reverse doubling condition, the author proves that [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] is bounded from Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) into Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) for p ∈ ( 1 , ∞ ) p\in \left(1,\infty ) and also bounded from spaces L 1 ( μ ) {L}^{1}\left(\mu ) into spaces L 1 , ∞ ( μ ) {L}^{1,\infty }\left(\mu ).
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In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.
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We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically calmost periodic functions and reconsider the notion of semi-c-periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent.
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For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces.
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This article proposes an inversion method for microwave tomography that combines a multifrequency data processing with a nonlinear regularization procedure formulated in Lebesgue spaces with nonconstant exponents.
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We prove the global well-posedness of the systems, as well as boundedness properties using the de Giorgi method and estimates in Lebesgue spaces.
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We demonstrate the richness of the class of MoE models by proving denseness results in Lebesgue spaces, when inputs and outputs variables are both compactly supported.
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We establish some analogs of Ulyanov’s and Andrienko’s Theorems on the embedding of the Hölder spaces of integrable functions on zero-dimensional second countable locally compact groups into the Lebesgue spaces or other Hölder spaces.
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