Real Interpolation and Two Variants of Gehring's Lemma

Acta Mathematica Academiae Scientiarum Hungaricae, 1955

2012

This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Periodica Mathematica Hungarica, 2020

A real valued functionfdefined on a real open intervalIis called$$\Phi $$Φ-monotone if, for all$$x,y\in I$$x,y∈Iwith$$x\le y$$x≤yit satisfies$$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$f(x)≤f(y)+Φ(y-x),where$$ \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+$$Φ:[0,ℓ(I)[→R+is a given nonnegative error function, where$$\ell (I)$$ℓ(I)denotes the length of the intervalI. Iffand$$-f$$-fare simultaneously$$\Phi $$Φ-monotone, thenfis said to be a$$\Phi $$Φ-Hölder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for$$\Phi $$Φ-monotonicity and$$\Phi $$Φ-Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper$$\Phi $$Φ-monotone and$$\Phi $$Φ-Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an ext...

Annali di Matematica Pura ed Applicata, 1982

In the present paper we study a general ]orm of Peetre's J-and K-methods o] interpolation. Special emphasis is given to the equivalence theorem for J-and K-spaces and to reiteration theorems.

2017

In this paper we are interested in polynomial interpolation of irregular functions namely those elements of L(R, μ) for μ a given probability measure. This is of course doesn’t make any sense unless for L functions that, at least, admit a continuous version. To characterize those functions we have, first, constructed, in an abstract fashion, a chain of Sobolev like subspaces of a given Hilbert space H0. Then we have proved that the chain of Sobolev like subspaces controls the existence of a continuous version for L functions and gives a pointwise polynomial approximation with a quite accurate error estimation.

Journal of Approximation Theory, 1969

Ukrainian Mathematical Journal

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.

2008

Journal of the London Mathematical Society, 1998

Journal of Approximation Theory, 1999

We estimate the distribution function of a Lagrange interpolation polynomial and deduce mean boundedness in Lp; p < 1:

This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman's inequality, which may be regarded as a discrete version of Pólya-Knopp's inequality and also as a natural limiting inequality of the discrete Hardy inequality. We present several simple proofs of and remarks (e.g. historical) about this inequality. In Chapter 3 we give some sharpenings and generalizations of Carleman's inequality. We discuss and comment on these results and put them into the frame presented in the previous chapter. In particular, we present some new proofs and results. In Chapter 4 we prove a multidimensional Sawyer duality formula for radially decreasing functions and with general weights. We also state the corresponding result for radially increasing functions. In particular, these results imply that we can describe mapping properties of operators defined on cones of such monotone functions. Moreover, we point out that these results can also be used to describe mapping properties of operators between some corresponding general weighted multidimensional Lebesgue spaces. In Chapter 5 we give a new weight characterization of the weighted Hardy inequality for decreasing functions and use this result to give a new weight characterization of the weighted Pólya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for characterizing the embedding Λ p (v) → Γ q (u) for the case 1 < p ≤ q < ∞. In Chapter 6 we make a unified approach to Hardy type inequalitits for decreasing functions and prove a result which covers both the Sinnamon result with one condition and Sawyer's result with two independent conditions for the case when one weight is nondecreasing. In all cases we point out that this condition is not unique and can even be chosen among some (infinite) scales of conditions. In Chapter 7 v vi Abstract we prove a weight characterization of L p ν [0, ∞)−L q μ [0, ∞

… instabilities in continuum mechanics (Edinburgh, 1985

Journal of Mathematical Analysis and Applications, 2018

Given a nondecreasing function f on [−1, 1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates "interpolatory estimates". In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f, •) of f evaluated at √ 1 − x 2 /n and were valid for all n ≥ 1. The current paper is devoted to proving that if f ∈ C r [−1, 1], r ≥ 1, then the interpolatory estimates are valid for the second modulus of smoothness of f (r) , however, only for n ≥ N with N = N(f, r), since it is known that such estimates are in general invalid with N independent of f. Given a number α > 0, we write α = r + β where r is a nonnegative integer and 0 < β ≤ 1, and denote by Lip * α the class of all functions f on [−1, 1] such that ω2(f (r) , t) = O(t β). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α ≥ 2 since 1985: If α > 0, then a function f is nondecreasing and in Lip * α, if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials Pn, of degree n, such that |f (x) − Pn(x)| ≤ C √ 1 − x 2 n α , x ∈ [−1, 1].

Nonlinear Analysis

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz-Zygmund space L a,r (Log L) β , is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 < a < ∞, β = 0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm. Contents 1 Introduction. Main results 2 2 Notations. Preliminary Lemmas 7 3 Computation of some K-functionals and characterization of the interpolation spaces (L p),α , L q),α ) θ,r 11 4 Small Lebesgue space as interpolation of usual Lebesgue spaces 22 5 Interpolation between small, Grand Lebesgue spaces and the associated K-functional 24 6 The critical case p = q. The interpolation space (L p) , L (p ) and its Kfunctional 28

Bulletin of the American Mathematical Society, 1974