Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient

Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.


Introduction
The Hardy-Littlewood maximal function was considered as a classical tool in various areas such as potential analysis and harmonic analysis from years and later in Sobolev space theory and partial differential equations see [1,4,12], and [13]. Sobolev space plays a significant role in dealing with existence and regularity of solutions of Partial Differential Equations see [5,8]. The Hardy-Littlewood maximal function bridging between functional analysis, sobolev spaces and partial differential equations [7,9,11,17]. Boundedness of Maximal function has been discussed earlier with different arguments such as, Boundedness and regularity of maximal functions on hardy-sobolev spaces discussed in [15]. Luiro [10,16] and [18]. With the strong arguments over the boundedness of Hardy-Lilltewood Maximal function, our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg sobolev inequality.
We start by recalling the definition of maximal function. Let , ∈ : | | is an open ball having center at ∈ and radius 0 then It shows the maximal function will essentially bounded with the boundedness of original function and thus finite everywhere. See [9] Hardy-Littlewood-Wiener theorem states that: This result indicates that maps from ' to weak ' and ∈ ' does not claim about ∈ ' and thus Hardy-Littlewood maximal operator is not bounded in ' . In this case we can get only weak type estimates. Hajłasz and Onninen raised same type of question in [15]. Later on Tanaka [6] gave its answer positively for 3 = 1.
This result shows that Hardy-Littlewood operator is bounded in ' for > 1 Lemma 2: If G ∈ H / , then for every ∈ I3 J , the 3 − 1 -dimensional measure of K 1,0 we have G = L ?ME ∫ > ?
In another way we can write for initial condition G R = 0 as From (3) and (4) This is the representation formula for a compactly supported continuously differential function in term of its gradient.
By Cauchy-Schwarz inequality and Lemma 2, we can write Where f denotes the Reisz Potential for g = 1.
For 0 < g < 3, Reisz Potential of order g can be deduced as One who interested in fundamental properties of Riesz potentials, see e.g. [12].
In case of compactly supported smooth functions, the above result is useful for pointwise bound of functions in term of Reisz potential of the gradient. Some authors have obtained some results for Reisz protentional for example Armin Schikorra and Daniel Spectory [2] established new ' -type estimate for Riesz potential. In [3], Petteri Harjulehto, Ritva Hurri-Syrjänen, obtained Pointwise estimates to the modified Riesz potential. Using the similar results, they obtained Poincare Inequality for irregular domain. The inequality of Gagliardo-Nirenberg-Sobolev type was established for nonisotropic Generalized Riesz Potential depending on λ−distance by Inan Cinar in [18]. Our point of interest here is to discuss boundedness of Reisz Potential by Maximal Operator and then to obtain Gagliardo-Nirenberg inequality Before moving to the main results, we shall elaborate few technical lemmas for Reisz potential for g 1 We can compute one part of product on right side as Since ∫`0 1, † 13a 3 ‡ = J ‡ , above inequality can be reduced as exponent in (11) can be expressed in the form It will help us to deduce the proof Sobolev Gagliardo-Nirenberg Inequality.