Optimal embeddings of generalized homogeneous Sobolev spaces

Abstract

We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in R n with K-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f .

The Riesz Potential Operator in Optimal Couples of Rearrangement Invariant Spaces

“…A direct approach to the same problem for the homogeneous Sobolev space with a norm |α|=k D α f E is used in [1] and similar results are proved. The problem of optimal embeddings of inhomogeneous Sobolev spaces, defined on a bounded domain in R n , is treated by somewhat different methods in [13,14,16,18,20,[22][23][24][25][26].…”

“…For example, if E = L r , 1 ≤ r < ∞, then the condition (8) means that s < n r . In the subcritical case and if β E < 1 we prove that the optimal target quasi- In the critical case we use real interpolation similarly to [11], but in a simpler way [1] and consider domain quasi-norms…”

Optimal Embeddings of Calderón Spaces in Hölder-Zygmund Spaces

“…A positive function b ∈ M + is said to be slowly varying on (1, ∞) if for all ε > 0 the function t ε b(t) is equivalent to an increasing function, and the function t −ε b(t) is equivalent to a decreasing function. By symmetry, we say that b is slowly varying on (0, 1) if the function t → b 1 t is slowly varying on (1, ∞). Finally, b is slowly varying if it is slowly varying on (0, 1) and (1, ∞).…”

“…During the last thirty years many authors have investigated the problem of optimal embeddings of Sobolev type spaces and of Besov type spaces. We refer, for example, to the monograph by Nikol'skij [38] and the papers by Hansson [24], Netrusov [37], Goldman [20], Kolyada [27], Edmunds and Triebel [16], Cwikel and Pustylnik [12], [13], Edmunds, Kerman and Pick [15], Maly and Pick [30], Milman and Pustylnik [35], Cianchi [9] or the more recent papers [23], [31], [26], [32], [18], [34], [33] and [1].…”

Optimal Regularity Properties of the Generalized Sobolev Spaces

“…The problem of the mapping properties of the Riesz potential in optimal couples of rearrangement invariant spaces is treated in [5][6][7]. The optimal embeddings of generalized Sobolev type spaces into rearrangement invariant spaces are characterized in several papers [5,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The characterization of the continuous embedding of the generalized Bessel potential spaces into the generalized Hölder-Zygmund spaces C , when is a weighted Lebesgue space, is given in [22].…”