Bochner–Riesz operators in grand lebesgue spaces

We provide the conditions for the boundedness of the Bochner–Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate for the constant appearing in the Lebesgue–Riesz norm estimation of the Bochner–Riesz operator and we investigate the convergence of the Bochner–Riesz approximation in Lebesgue–Riesz spaces.

The classical Lebesgue-Riesz norm f p , p ∈ [1, ∞], for the function f , is denoted by and the corresponding Banach space is, as usually, Denote, for an arbitrary linear or quasi linear operator U , acting from L p to L q , p, q ≥ 1, its norm Of course, the operator U is bounded as the operator acting from the space L p into the space L q iff U p,q < ∞. More generally, for the operator U acting from some Banach space F equipped with the norm ⋅ F into another (in the general case) Banach space D, having the norm ⋅ D , denote as usually There exists a huge numbers of works devoted to the p, q estimates for Bochner-Riesz operators B α R , as a rule for the case q = p, see e.g. [21, chapter 5], [7,8,11,26,31,38,39], etc.
Our aim, in this paper, is to extend some results contained in the above mentioned works concerning upper estimates for the norm of the Bochner-Riesz operator, in the case of different Lebesgue-Riesz spaces (Section 2) and to extend these estimates to the so-called Grand Lebesgue Spaces (GLS), see Section 3.
We deduce also a non trivial quantitative lower estimate for the coefficient in the Lebesgue-Riesz norm estimation for the Bochner-Riesz operator (Section 4) and we study the convergence of the Bochner-Riesz approximation in Lebesgue-Riesz spaces (Section 5).
2 Norm estimates for the Bochner-Riesz operator acting between different Lebesgue-Riesz spaces We clarify slightly the known results about the Lebesgue-Riesz p, r norms for the operators B α R defined in (1.1), see e.g. [8,21,10,26,39], ecc. It is important to observe that the Bochner-Riesz operator B α R may be rewritten as a convolution, namely is the Bessel function of order λ and z = (z, z). Notice that, under our restriction on α, Evidently, as long as R > 0.
Assume here and in the sequel and introduce the value so that where the inequality on the left hand side is true due to the restriction α > −1.
Moreover, using the well-known results about the behavior of the Bessel functions (see, e.g., [8, p. 172 it is easy to estimate and K λ q = ∞ otherwise (see also, e.g., [21, pp. 339-341]). Correspondingly, Furthermore, we recall the Beckner-Brascamp-Lieb-Young inequality for the convolution (see [3,4]) where p, q, r ≥ 1 and Recently in [20] has been given a generalization of the convolution inequality in the context of the Grand Lebesgue Spaces (see Section 3 for the definition), built on a unimodular locally compact topological group.
The estimate (2.10) is essentially non-improbable. Indeed, the equality in (2.10) is attained iff both functions f, g are proportional to Gaussian densities, namely there exists positive constants c 1 , c 2 , c 3 , c 4 such that where ⋅ is the euclidean norm. Then the convolution f * g is also Gaussian and The following result about the boundedness of the Bochner-Riesz operator, acting between different Lebesgue-Riesz spaces, holds. and (2.14) Let q, r ≥ 1 such that 1 + 1 r = 1 q + 1 p and assume q > q 0 , r > r 0 , p ≤ p 0 , where Then the Bochner-Riesz operator satisfies Proof.
If f ∈ L p for some value p > 1, then by (2.10) it follows where 1 + 1 r = 1 q + 1 p and p, q, r ≥ 1, q > q 0 . On the other words, Moreover it is easy to verify that r 0 , defined in (2.15), is such that r 0 > p. Under our restrictions W (α, n, R; p, r) is finite and positive. So we conclude that estimate (2.18) holds. ✷

Main result. Boundedness of Bochner-Riesz operators in Grand Lebesgue Spaces (GLS)
We recall here, for reader convenience, some known definitions and facts from the theory of Grand Lebesgue Spaces (GLS). For instance are generating functions. If where C ∞ ∶= 0, C ∈ R (extremal case), then the corresponding Gψ space coincides with the classical Lebesgue-Riesz space L r = L r (R d ).
These spaces are rearrangement invariant (r.i.) Banach functional (complete) spaces; their fundamental functions have been considered in [35]. They do not coincide, in the general case, with the classical Banach rearrangement functional spaces: Orlicz, Lorentz, Marcinkiewicz etc., see [30,33]. The belonging of a function f ∶ R n → R to some Gψ space is closely related with its tail function behavior as t → 0+ as well as when t → ∞, see [27,29].
The Grand Lebesgue Spaces can be considered not only on the Euclidean space R n equipped with the Lebesgue measure, but also on an arbitrary measurable space with sigma-finite non-trivial measure.
The proof is simple and alike as the one in [36]. First we observe that ν(r) is finite. One can assume, without loss of generality, f Gψ = 1, then f p ≤ ψ(p), p ∈ (a, b). Applying the inequality (2.18) we have Taking the minimum over p subject to our limitations, we get [ W (α, R; p, r) ⋅ ψ(p) ] = C(α, n, R) ν(r) = C(α, n, R) ν(r) f Gψ , (3.8) which is quite equivalent to our claim in (3.6). ✷ 4 Lower bound for the coefficient in the Lebesgue-Riesz norm estimate for the Bochner-Riesz operator.
Let p, r > 1, n > 1 and let us introduce the following variable where W (α, n, R; p, r) is defined in Section 2. Our target in this Section is a lower bound for the above variable. Then Q n (p, r) ≥ Θ n, pr pr + p − r , r > p. Remark 4.1. The possible case when Q n (p, r) = +∞ can not be excluded.

Proof.
We will apply equality (2.12), in which we choose the ordinary Gaussian density and take α = R 2 2. Obviously Q n (p, r) ≥ lim R→∞ W (R 2 2, n, R; p, r).
We have where I(A) denotes the indicator function of the (measurable) set A, A ⊂ R n . Therefore, as R → ∞ , by virtue of dominated convergence theorem.
If we take f = f 0 , then in (4.2) the convolution of two Gaussian densities appears. It remains to apply the relation (2.12); we omit some simple calculations. ✷

Convergence of Bochner-Riesz operators.
We investigate here the convergence, as R → ∞, of the family of Bochner-Riesz approximations B α R [f ] to the source function f in the Lebesgue-Riesz norm L p (R n ), p ∈ (1, ∞), in addition to the similar results in [21,11,26], etc.
For any function f ∈ L p (R n ), its modulus of L p continuity is defined alike as in approximation theory [1, chapter V] : We apply now the triangle inequality for the L p norm in the integral form Note that, under the above conditions, so that (5.2) follows again from the dominated convergence theorem. ✷ Remark 5.1. As a slight consequence, under the above conditions, if f ∈ L p (R n ), then and, consequently, The case p = ∞ requires a separate consideration. Introduce the Banach space C 0 (R n ) as a collection of all bounded and uniformly continuous functions f ∶ R n → R , equipped with the ordinary norm, As above 3) The assertion of Theorem 5.1 under the same conditions remains true in the case p = ∞.
Theorem 5.2. Under the same conditions of Theorem 5.1, for any function f ∶ R n → R from the space C 0 (R n ) , its Bochner-Riesz approximation B α R converges uniformly to the source function f , that is Proof. The proof is the same as in Theorem 5.1 and may be omitted. ✷ Remark 5.2. As before, if f ∈ C 0 (R n ), then 6 Concluding remarks.
A. In our opinion, the method described in this paper may be essentially generalized on more operators of convolutions type, linear or not. See some preliminary results [30].
B. It is interesting to generalize the estimates obtained in the previous Sections to the so-called maximal operators associated with the Bochner-Riesz one considered here, in the spirit of the works [16,25], and so one: