By Jan Lang

The major subject matter of the ebook is the examine, from the perspective of s-numbers, of imperative operators of Hardy kind and comparable Sobolev embeddings. within the idea of s-numbers the assumption is to connect to each bounded linear map among Banach areas a monotone lowering series of non-negative numbers so one can the class of operators based on the best way those numbers procedure a restrict: approximation numbers offer a particularly very important instance of such numbers. The asymptotic habit of the s-numbers of Hardy operators performing among Lebesgue areas is set right here in a large choice of circumstances. The evidence equipment contain the geometry of Banach areas and generalized trigonometric services; there are connections with the speculation of the p-Laplacian.

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**Example text**

32) we obtain the estimate |τm (p)| ≤ 4π p /(π m)2 (m odd). 34) Next we consider the dependence of sin p (nπ pt) on p. 5. Suppose that 1 < p < q < ∞. Then the function f defined by f (t) = is strictly decreasing on (0, 1). 2 Basis Properties 45 Proof. Let g(t) = (1 − t q )1/q (0 < t < 1). (1 − t p )1/p For all t ∈ (0, 1), t p−1 −t q−1 + q 1−t 1−tp g (t) = g(t) Put = (t p − t q)g(t) > 0. t(1 − t q )(1 − t p) −1 G(t) = sin−1 p (t) − g(t) sinq (t) and observe that G (t) = −(sin−1 q t)g (t) < 0 in (0, 1).

Our choice was p p largely motivated by the wish to have available the identity sin p x + cos p x = 1, while other authors attached greater importance to different properties. Power series expansions for his versions of sin p , cos p and tan p are given by Linqvist [90]; see also the detailed work in this direction on related functions by Peetre [104]. g. for sin p (x + y)) seem to be known. Further details of properties of p-trigonometric functions are given in [20]. 2. The only work on the basis properties of the sin p functions of which we are aware is that of [9].

1 − (sin p θ ) p s p )−1/p ds ≤ πp 2 for all θ ∈ (0, π p /2], the result follows. 2. For all p ∈ (1, ∞) and all t ∈ (0, 1/2), sin p (π pt) > 2t. Proof. 6, sin p θ > 2θ /π p if 0 < θ < π p /2. Now put θ = π pt. Given any function f on [0, 1], we extend it to a function f on R+ := [0, ∞) by setting f (t) = − f (2k − t) for t ∈ [k, k + 1], k ∈ N. 35) With this understanding, we define maps Mm : Lq (0, 1) → Lq (0, 1) (1 < q < ∞) by Mm g(t) = g(mt), m ∈ N, t ∈ (0, 1). 36) Note that Mm en = emn . 1.