Note on the Equivalence of Special Norms on the Lebesgue Space

Round 1

Reviewer 1 Report

I found the paper interesting since it develops a novel approach for the compact embedding in Lebesgue spaces. Especially establishing the equivalence of the norm generated by the infinitesimal generator of the shift semigroup in a direction and the norm of the Nicodemus space is an original approach. The paper should be considered for the publication in Axioms.

The authors are strongly encouraged to provide some applications of these norm equivalencies in Lebesgue spaces, especially in variational setting. In this regard, I suggest that the following papers should be benefited as well as added to the references.

1-https://doi.org/10.1006/jmaa.2000.7617

2-https://doi.org/10.1016/j.jmaa.2011.12.029

Author Response

I am sincerely grateful to you for your appreciation of my results, I found the  papers, you so kindly  pointed  out to me, worth to be cited it the context of the introduction section. I was previously  acquainted with this topic a little, and I   think they (the papers) should be studied  properly to  find a possible  approach to generalise and  apply afterwards  my own results  in  the theory of Lebesgue spaces with the variable exponent.

Author Response File: Author Response.pdf

Reviewer 2 Report

Dear authors, please see attached my review.

Comments for author File: Comments.pdf

Author Response

I am sincerely grateful to  you  for a number of  valuable remarks. However, let us consider them carefully and consistently.  Below, I attached the pdf file. 

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The this paper is about a norm based on an infinitesimal generator. A representation of a fractional integro-differential operator is used. The equivalence of norms in functional spaces is attended. 

Definitions are well defined by using a precise explanation of the symbols.

The result in Corollary 2 presents an interesting equivalence, which could applied in a more specific way to the embedding kind result, so useful also in algebraic geometry.

Theorem 2 well use all the methods defined inside the work, but, anew, the definition contain results which are similar to existing theories about  the involved norms and operators.

Major remarks:

• Please better specify the novel or the existing results.
• Introduction must be enriched by giving more references technically linked to the topics, the most of the section is about discursive results.
• i can't consider Lemma 1 as a novel result, the estimate is note in literature by considering, also, different kinds of order, the proof consider usual methods for this kind of result.
• About Lemma 2, this is a result already used in similar way in numerical analysis, for instance. 
• Lemma 3 reduces to the analysis of the matrix properties by using trivial algebraic calculus methods, I consider the work devoted to more constructed analytical results.
• Theorem 2: the equivalence it's completely about a general note results not so specifically linked to the paper's main topic.
• The using of integro-differential operator is fictitious, here are never really used and taken into account the inner properties and the methods regarding its application. 
• Conclusions reflect my the most of my major remarks.
• A final example is important to be added.

Minor remarks:

• Page 1: write "m-accretive" instead of "m-accretive" (and on).
• The "Uniformly elliptic operator in the divergent form" could be inserted in a proper section or subsection.

Author Response

I am sincerely grateful to  you  for a number of  valuable remarks. However, let us consider them carefully and consistently. Below, I attached pdf file. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper ``One more theorem on norm equivalence in the
Lebesgue space''
Maksim V. Kukushkin is very qualitative and can be published in Axioms.

Author Response

Enormous gratitude for your believe in my results and many thanks for your attention. Sincerely yours Maksim! 

Reviewer 3 Report

The paper is devoted to studying a norm constructed by the action of an n-parametric semigroup which is generated by an infinitesimal operator on the Hilbert space of square-integrable functions on a domain in R^n. The author proved that this norm is equivalent to the original norm of the space. The result is interesting and relevant. However, the paper contains a lot of language mistakes and unclear places. So, I recommend a substantial revision.

I upload the list of comments in the attachment.

Comments for author File: Comments.pdf

Author Response

I am sincerely grateful to  you  for a number of  valuable remarks. However, let us consider them carefully and consistently. Below, I attached a pdf file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I can't find relevant novelties respect the previous version.

Reviewer 3 Report

The paper looks better but it still not readable and the style is not friendly for readers. Axioms is not a specialized journal and articles should be self-enough with complete and clear explanations.   More comments are attached.

Comments for author File: Comments.pdf