Optimal Weak Type Estimates for the P-ADIC Hardy Type Operators on Higher-Dimensional Product Spaces

In recent years, p-adic analysis has been widely used in quantum mechanics, the probability theory, and the dynamical systems [1, 2]. Meanwhile, there is an increasing attention in pseudo-dierential equations, wavelet theory, and harmonic analysis (see [3–8]). For a prime number p, let Qp be the eld of p-adic numbers, a nonzero rational number x is represented as x pm/n, where c is an integer and the integers m, n are not divisible by p. en, the norm is dened as |x|p p− , and it is easy to see that the norm satises the following properties:

For a prime number p, let Q p be the eld of p-adic numbers, a nonzero rational number x is represented as x p c m/n, where c is an integer and the integers m, n are not divisible by p. en, the norm is de ned as |x| p p − c , and it is easy to see that the norm satis es the following properties: (i) |x| p ≥ 0, ∀x ∈ Q p , |x| p 0⇔x 0 (ii) |xy| p |x| p |y| p , ∀x, y ∈ Q p (iii) |x + y| p ≤ max |x| p , |y| p , ∀x, y ∈ Q p , in the case when |x| p ≠ |y| p , we have |x + y| p max |x| p , |y| p It is well known that Q p is a typical model of non-Archimedean local elds. From the standard p-adic analysis, any x ∈ Q p / 0 { } can be uniquely represented as a canonical form where α k , c ∈ Z, α 0 ≠ 0 ≤ α k < p, note that the series (1) converges with respect to the norm |x| p because one has |p c α k p k | p p − c− k . e space Q n p consists of elements x (x 1 , x 2 , . . . , x n ), where x i ∈ Q p , i 1, 2, . . . , n. e norm in this space is e symbols B c (a) and S c (a) represent, respectively, the ball and the sphere with center at a ∈ Q n p and radius p c , de ned by As Q n p is a locally compact commutative group with respect to addition, there exists a Harr measure dx on Q n p , which is unique up to a positive constant factor and is translation invariant, that is, d(x + a) dx. We normalize the measure dx such that where |B| H denotes the Harr measure of a measure subset B of Q n p . By simple calculation, we can obtain that B c (a) was introduced by Hardy in [9], and a celebrated integral inequality states that It was also shown that the constant factor q/(q − 1) is optimal, knowing its importance in analysis.
Faris in [10] and Christ and Grafakos in [11] proposed an extension of (1) and its adjoint to the n-dimensional Euclidian spaces R n of which the equivalent forms are e norm of H and H * on L q (R n ) was evaluated and found to be equal to that of the classical Hardy operator. For more details about the boundedness of the Hardy operator and its adjoint, we included some references [12][13][14].
In 2020, Li et al. [27] introduced the definition of the fractional Hardy operator on higher-dimensional product spaces as follows: where f be a nonnegative integrable function on Furthermore, the corresponding operator norm on the weak Lebesgue product spaces was obtained.
Next, we will introduce the definition of the fractional Hardy operator on higher-dimensional p-adic product spaces and obtain sharp weak bounds.
Define the fractional p-adic Hardy operator on higher-dimensional product spaces by In 2020, Wang et al. [28] gave the definition of fractional conjugate Hardy operator on higher-dimensional product spaces as follows: where f be a nonnegative integrable function on , and they also got the corresponding operator norm on the weak Lebesgue product spaces.
Next, we will give a higher-dimensional version of the fractional p-adic conjugate Hardy operator and obtain sharp weak bounds.
Define the fractional p-adic conjugate Hardy operator on higher-dimensional product spaces by In this article, we will obtain sharp weak bounds for the fractional p-adic Hardy operators and its conjugate operators on the p-adic Lebesgue product spaces. Our method of proving the main results involves a frequent use of the following formula:

Sharp Weak Bounds for Fractional Hardy Operators
is section considers the problem of obtaining optimal weak bounds for H p β 1 ,...βm and our results as follows.
To obtain the desired result, we need the following lemma.