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We obtain pointwise and integral type estimates of higher-order partial moduli of continuity in C via partial derivatives. Also, a Gagliardo–Nirenberg type inequality for partial derivatives in a fixed direction is proved.
Our methods enable us to study the case when different partial derivatives belong to different spaces, including the space
L
1.
KeywordsModuli of continuity–Gagliardo–Nirenberg type inequalities–Sobolev spaces–Besov norms–Embeddings–Rearrangements

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... By (2.6), it is sufficient to obtain (2.7) in the case s > 0; for this case, see [11], [12] - [15], and references therein. We recall also the thermic definition of Triebel-Lizorkin spaces. ...

... The following lemma is a slight modification of Lemma 2.4 in [15]. ...

We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the
function by Sobolev norms and homogeneous Besov quasinorms with negative
smoothness. We prove also other versions involving Besov or Triebel-Lizorkin
quasinorms. These inequalities can be considered as refinements of Sobolev type
embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities
in some limiting cases. Our methods are based on estimates of rearrangements in
terms of heat kernels. These methods enable us to cover also the case of
Sobolev norms with p = 1.

... where a constant C is independent of δ and ω. Nowadays the Ulyanov inequalities are used in approximation theory, function spaces, and interpolation theory (see, e.g., [7,15,19,31,32,37,48,49,51,52,53,66,76,87,99,104]). Very recently [31], sharp Ulyanov inequalities were shown between the Lorentz-Zygmund spaces L p,r (log L) α−γ , α, γ > 0, and L q,s (log L) α over T d in the case 1 < p ≤ q < ∞, and the corresponding embedding results were established. ...

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ω α ( f , t ) q \omega _\alpha (f,t)_q and ω β ( f , t ) p \omega _\beta (f,t)_p for 0 > p > q ≤ ∞ 0>p>q\le \infty . A similar problem for the generalized K K -functionals and their realizations between the couples ( L p , W p ψ ) (L_p, W_p^\psi ) and ( L q , W q φ ) (L_q, W_q^\varphi ) is also solved.
The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity sup T n ‖ D ( ψ ) ( T n ) ‖ q ‖ D ( φ ) ( T n ) ‖ p , 0 > p > q ≤ ∞ , \begin{equation*} \sup _{T_n} \frac {\Vert \mathcal {D}(\psi )(T_n)\Vert _q}{\Vert \mathcal {D}({\varphi })(T_n)\Vert _p},\qquad 0>p>q\le \infty , \end{equation*} where the supremum is taken over all nontrivial trigonometric polynomials T n T_n of degree at most n n and D ( ψ ) , D ( φ ) \mathcal {D}(\psi ), \mathcal {D}({\varphi }) are the Weyl-type differentiation operators.
We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

... [60]). In particular, it can be proved in the same way as Lemma 2.4 in [35]. Lemma 6.2. ...

The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis.
This thesis is devoted to the investigation of various properties of functions of generalized bounded variation. In particular, we obtain the following results:
- sharp relations between spaces of generalized bounded variation and spaces of functions defined by integral smoothness conditions (e.g., Sobolev and Besov spaces);
- optimal properties of certain scales of function spaces of frac- tional smoothness generated by functionals of variational type;
- sharp embeddings within the scale of spaces of functions of bounded p-variation;
- results concerning bivariate functions of bounded p-variation, in particular sharp estimates of total variation in terms of the mixed Lp-modulus of continuity, and Fubini-type properties.

... To prove Theorem 4.1, we first use a variant of estimate (1.1), proved in [23], which states that if n, k ∈ N, k ≤ n, and f is a function such that the norm of its distributional gradient |∇ k f | belongs locally to the Lorentz space L n/k,1 (R n ), then f can be redefined on a set of measure zero so that f is continuous on R n and the k-modulus of smoothness ω k ( f, ·) satisfies ...

We use an estimate of the k-modulus of smoothness of a function f such that the norm of its distributional gradient |∇ k f | belongs locally to the Lorentz space L n/k,1 (R n), k ∈ N, k ≤ n, and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces X (R n). Target spaces of our embed-dings are generalized Hölder spaces defined by means of the k-modulus of smoothness (k ∈ N). General results are illustrated with examples. Keywords Rearrangement-invariant Banach function space · Modulus of smoothness · Distributional gradient · Lorentz space · Sobolev-type space · Banach lattice · Hölder-type space · Embeddings

... [12]). In particular, it can be proved in the same way as Lemma 2.4 in [7]. ...

We obtain a necessary and sufficient condition for embeddings of integral
Lipschitz classes Lip(\alpha; p) into classes \Lambda BV of functions of
bounded \Lambda-variation.

... Note also that we have proved an analogous result for homogeneous Sobolev spaces ˙ W k+1,n/k (R n ) defined as the set of functions on R n such that the norm of its distributional gradient |∇ k+1 f | belongs to L n/k (R n ) (see, again, Example 5.11 below). To prove Theorem 4.1, we first use a variant of estimate (1.1), proved in [23], which states that if n, k ∈ N, k ≤ n, and f is a function such that the norm of its distributional gradient |∇ k f | belongs locally to the Lorentz space L n/k,1 (R n ), then f can be redefined on a set of measure zero so that f is continuous on R n and the k-modulus of smoothness ω k ( f, ·) satisfies ...

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H
σ
X(IR
n
) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IR
n
), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR
n
) is the Lorentz-Karamata space Lp,q;b(I Rn)L_{p,q;b}({{\rm I\kern-.17em R}}^n). In particular, we are able to characterize optimal embeddings of Bessel potential spaces HsLp,q;b(I Rn)H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield
new and sharp embeddings of Sobolev-Orlicz spaces W
k + 1
L
n/k
(logL)
α
(IR
n
) and W
k
L
n/k
(logL)
α
(IR
n
) into generalized Hölder spaces.
KeywordsSlowly varying functions-Lorentz-Karamata spaces-Rearrangement-invariant Banach function spaces-Bessel potentials-(fractional) Sobolev-type spaces-Hölder-type spaces-Zygmund-type spaces-Embedding theorems
Mathematics Subject Classifications (2000)46E35-46E30-26B35-26A12-26D15-26A15-26A16-26B35-47B38-26D10

We study the Sobolev embedding in $\cdot$ subcritical case, that is, $(W^k_p(\Omega))_0 \hookrightarrow L_{p^\ast,p}(\Omega)$ for $k < d/p,$ $\cdot$ critical case, that is, $(W^{k}_p(\Omega))_0 \hookrightarrow Y$ for $k = d/p$ and appropriate $Y,$ $\cdot$ and supercritical case, that is, $\dot{W}^{k}_p(\mathcal{X}) \hookrightarrow Y$ for $k > d/p, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d\}$ and appropriate $Y.$ We obtain characterizations of these embeddings in terms of pointwise inequalities involving rearrangements and moduli of smoothness/derivatives of functions and via extrapolation theorems for corresponding smooth function spaces. Applications include, among others, Ulyanov--Kolyada type inequalities for rearrangements, inequalities for moduli of smoothness, sharp Jawerth--Franke embeddings for Lorentz--Sobolev spaces, various characterizations of Gagliardo--Nirenberg, Trudinger, Maz'ya--Hansson--Br\'ezis--Wainger and Br\'ezis--Wainger embeddings.

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from $L_p$, $1<p<\infty$. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from Lp, 1 < p < ∞. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.

The paper deals with Gagliardo-Nirenberg inequalities in function spaces of type B
p,q
s
(ℝn
) and F
p,q
s
(ℝn
).

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of a one-sided nonlocal boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients. We obtain an estimate for the solution of this problem in the corresponding spaces.

From the Preface:
"Three classical interpolation theorems form the foundation of the modern theory of interpolation of operators. They are the M. Riesz convexity theorme (1926), G.O. Thorin’s complex version of Riesz’ theorem (1939), and the J. Marcinkiewicz interpolation theorem (1939). The ideas of Thorin and Marcinkiewicz were reworked some twenty years later into an abstract theory of interpolation of operators on Banach spaces and more general topological spaces. Thorin’s technique has given rise to what is now know as the complex method of interpolation, and Marcinkiewicz’ to the real method. Both have found widespread application, have extensive literatures attached to them, and remain very much alive as subjects of current research.
This is a book primarily about the real method of interpolation. Our goal has been to motivate and develop the entire theory from its classical origins, that is, through the theory of spaces of measurable functions. Although the influence of Riesz, Thorin, and Marcinkiewicz is everywhere evident, the work of G.H. Hardy, J.E. Littlewood, and G. Polya on rearrangements of functions also plays a seminal role. It is through the Hardy-Littlewood-Polya relation that spaces of measurable functions and interpolation of operators come together, in a simple blend which has the capacity for great generalization. Interpolation between the Lebesgue spaces L<sup>1</sup> and L<sup>∞</sup> is thus the prototype for interpolation between more general pairs of Banach spaces. This theme airs constantly throughout this book.
The theory and applications of interpolation are as diverse as language itself. Our goal is not a dictionary, or an encyclopedia, but instead a brief biography of interpolation, with a beginning and an end, and (like interpolation itself), some substance in between.
The book should be accessible to anyone familiar with the fundamentals of real analysis, measure theory, and functional analysis. The standard advanced, undergraduate of beginning graduate courses in these disciplines should suffice. The exposition is essentially self-contained. "

This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book "Fractals and Spectra" (MMA 91). It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated.

Let E be a Banach space. Let L (1) 1 (ℝ d ,E) be the Sobolev space of E-valued functions on ℝ d with the norm ∫ ℝ d ∥f∥ E dx+∫ ℝ d ∥∇f∥ E dx=∥f∥ 1 +∥∇f∥ 1 · It is proved that if f∈L (1) 1 (ℝ d ,E) then there exists a sequence (g m )⊂L (1) 1 (ℝ d ,E) such that f=∑ m g m ; ∑ m (∥g m ∥ 1 +∥∇g m ∥ 1 )<∞; and ∥g m ∥ ∞ 1/d ∥g m ∥ 1 (d-1)/d ≤b∥∇g m ∥ 1 for m=1,2,⋯, where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding L (1) 1 (ℝ d ,E)↪L 2 (ℝ d ,E). In particular, the embedding into Besov spaces L (1) 1 (ℝ d ,E)↪B p,1 θ(p,d) (ℝ d ,E) is proved, where θ(p,d)=d(p -1 +d -1 -1) for 1<p≤d/(d-1), d=1,2,⋯. The latter embedding in the scalar case is due to Bourgain and Kolyada.

In this paper a study is made of multiplicative inequalities of Gagliardo-Nirenberg type that connect partial moduli of continuity and partial derivatives of functions with respect to a fixed variable in different Lorentz norms. The main results are expressed by estimates of the form
where ,
and the exponents and satisfy certain conditions. In particular, these estimates imply optimal inequalities involving Besov norms and Lorentz norms. The limit case and estimates in terms of total variation are also studied.

We investigate the spaces of functions on R n for which the gene-ralized partial derivatives D r k k f exist and belong to different Lorentz spaces L p k ,s k . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non-increasing rearrangements. These methods enable us to cover also the case when some of p k 's are equal to 1.

Incluye bibliografía e índice

Integral Representation of Functions and Imbedding Theo-rems, vols

- O V Besov
- V P Il 'in

Besov, O.V., Il'in, V.P., Nikol'ski˘ ı, S.M.: Integral Representation of Functions and Imbedding Theo-rems, vols. 1–2.