# zbMATH — the first resource for mathematics

Tractable embeddings of Besov spaces into small Lebesgue spaces. (English) Zbl 1361.46026
If $$1\leq p<n$$, then $\Big( \int^1_0 (1 - \log t)^{p/2} f^* (t)^p \, dt \Big)^{1/p} \leq c \big( \| f \, | L^p \| + \| \nabla f\, | L^p \| \big),$ $$f \in W^{1,p}(\mathbb R^n)$$, $$\text{supp }f \subset [0,1]^n$$, where $$c>0$$ is independent of the dimension $$n$$. Here, $$f^*$$ is the usual decreasing rearrangement of $$f$$. There are counterparts with Besov spaces $$B^s_{p,q}$$ in place of the Sobolev spaces $$W^{1,p}$$. The main point is the control of the related constants $$c$$ on the dimension $$n$$. Assertions of this type are called tractable (or dimension-controllable). The paper contributes to this topic replacing the left-hand side of the above inequality by the so-called small Lebesgue spaces, normed by $\| f \, | L^{(p,b,q} (\mathbb T^n) \| = \Big( \int^1_0 \Big[ (1- \log t)^b \Big( \int^t_0 f^* (s)^p \, ds \Big)^{1/p} \Big]^q \frac{dt}{t} \Big)^{1/q}\,,$ where $$\mathbb T^n = [0,1]^n$$ is the $$n$$-torus. The arguments are based on periodic Besov spaces and related approximations, (limiting) interpolations and extrapolations.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42A10 Trigonometric approximation 46B70 Interpolation between normed linear spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text:
##### References:
 [1] Bennett, Intermediate spaces and the class Llog+L, Ark. Mat. 11 pp 215– (1973) · Zbl 0266.46025 · doi:10.1007/BF02388518 [2] Bennett, On Lorentz-Zygmund spaces, Dissertationes Math. 175 pp 1– (1980) [3] Bennett, Interpolation of Operators (1988) · Zbl 0647.46057 [4] Bergh, Interpolation Spaces. An Introduction (1976) · doi:10.1007/978-3-642-66451-9 [5] Cobos, Embeddings of Besov spaces of logarithmic smoothness, Studia Math. 223 pp 193– (2014) · Zbl 1325.46039 · doi:10.4064/sm223-3-1 [6] Cobos, Approximation spaces limiting interpolation and Besov spaces, J. Approx. Theory 189 pp 43– (2015) · Zbl 1326.46020 · doi:10.1016/j.jat.2014.09.002 [7] Cobos, On Besov spaces of logarithmic smoothness and Lipschitz spaces, J. Math. Anal. Appl. 425 pp 71– (2015) · Zbl 1320.46028 · doi:10.1016/j.jmaa.2014.12.034 [8] Cobos, On an extreme class of real interpolation spaces, J. Funct. Anal. 56 pp 2321– (2009) · Zbl 1211.46020 · doi:10.1016/j.jfa.2008.12.013 [9] Cobos, Equivalence of K- and J-methods for limiting real interpolation spaces, J. Funct. Anal. 261 pp 3696– (2011) · Zbl 1251.46008 · doi:10.1016/j.jfa.2011.08.018 [10] Cobos, On a limit class of approximation spaces, Numer. Funct. Anal. Optim. 11 pp 11– (1990) · Zbl 0729.41033 · doi:10.1080/01630569008816358 [11] Cobos, Description of logarithmic interpolation spaces by means of the J-functional and applications, J. Funct. Anal. 268 pp 2906– (2015) · Zbl 1335.46018 · doi:10.1016/j.jfa.2015.03.012 [12] DeVore, Constructive Approximation (1993) · doi:10.1007/978-3-662-02888-9 [13] DeVore, Weak interpolation in Banach spaces, J. Funct. Anal. 33 pp 58– (1979) · Zbl 0433.46062 · doi:10.1016/0022-1236(79)90018-1 [14] Fratta, A direct approach to the duality of grand and small Lebesgue spaces, Nonlinear Anal. 70 pp 2582– (2009) · Zbl 1184.46032 · doi:10.1016/j.na.2008.03.044 [15] Edmunds, Hardy Operators, Function Spaces and Embeddings (2004) · Zbl 1286.46001 · doi:10.1007/978-3-662-07731-3 [16] Fehér, On an extremal scale of approximation spaces, J. Comput. Anal. Appl. 3 pp 95– (2001) · Zbl 1033.46021 [17] Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math. 51 pp 131– (2000) · Zbl 0960.46022 [18] Fiorenza, Grand and small Lebesgue spaces and their analogs, Z. Anal. Anwend. 23 pp 657– (2004) · Zbl 1071.46023 · doi:10.4171/ZAA/1215 [19] Fiorenza, An improvement of dimension-free Sobolev imbeddings in r.i. spaces, J. Funct. Anal. 267 pp 243– (2014) · Zbl 1312.46043 · doi:10.1016/j.jfa.2014.04.011 [20] Gomez, Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc. 34 pp 305– (1986) · Zbl 0644.42014 · doi:10.1112/jlms/s2-34.2.305 [21] Haroske, Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers, J. Approx. Theory 128 pp 151– (2004) · Zbl 1055.46020 · doi:10.1016/j.jat.2004.04.008 [22] Iwaniec, On the integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. Anal. 119 pp 129– (1992) · Zbl 0766.46016 · doi:10.1007/BF00375119 [23] Jawerth, Extrapolation Theory with Applications, Memoirs of the American Mathematical Society 440 (1991) · Zbl 0733.46040 [24] Karadzhov, Extrapolation theory: new results and applications, J. Approx. Theory 133 pp 38– (2005) · Zbl 1081.46018 · doi:10.1016/j.jat.2004.12.003 [25] Kolyada, On limiting embeddings of Besov spaces, Studia Math. 171 pp 1– (2005) · Zbl 1090.46026 · doi:10.4064/sm171-1-1 [26] Krbec, On dimension-free Sobolev imbeddings I, J. Math. Anal. Appl. 387 pp 114– (2012) · Zbl 1238.46027 · doi:10.1016/j.jmaa.2011.08.061 [27] Krbec, On dimension-free Sobolev imbeddings II, Rev. Mat. Complut. 25 pp 247– (2012) · Zbl 1280.46023 · doi:10.1007/s13163-011-0068-5 [28] Martín, Pointwise symmetrization inequalities for Sobolev functions and applications, Adv. Math. 225 pp 121– (2010) · Zbl 1216.46026 · doi:10.1016/j.aim.2010.02.022 [29] Milman, Lecture Notes in Mathematics (1994) [30] Nikolski, Approximation of Functions of Several Variables and Imbedding Theorems (1975) · doi:10.1007/978-3-642-65711-5 [31] Novak, EMS Tracts in Mathematics Vol. 6 (2009) [32] Novak, EMS Tracts in Mathematics Vol. 12 (2010) [33] Petrushev, Rational Approximation of Real Functions Enciclopedia of Mathematics and Its Applications (1987) · Zbl 0644.41010 [34] Pietsch, Approximation spaces, J. Approx. Theory 32 pp 115– (1981) · Zbl 0489.47008 · doi:10.1016/0021-9045(81)90109-X [35] Schmeisser, Topics in Fourier Analysis and Function Spaces (1987) · Zbl 0661.46024 [36] Temlyakov, Approximation of Periodic Functions (1994) · Zbl 0899.41001 [37] Triebel, Interpolation Theory, Function Spaces, Differential Operators (1978) · Zbl 0387.46033 [38] Triebel, Proc. Function Spaces IX, Banach Center Publ pp 361– (2011) [39] Triebel, Proc. Function Spaces X, Banach Center Publ pp 229– (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.